Stewart 8th §6.1: Areas Between Curves

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Stewart 8th §6.1: Areas Between Curves 0/68
1 Areas Between Curves · Level 2
Find the area of the shaded region. [Figure: \(y = \sqrt[4]{x}\), \(y = \dfrac{1}{x}\), from \(x = 1\) to \(x = 8\)]
2 Areas Between Curves · Level 2
Find the area of the shaded region. [Figure: \(y = e^x\), \(y = x e^{x^2}\), from \(x = 0\) to the intersection near \((1, e)\)]
3 Areas Between Curves · Level 2
Find the area of the shaded region. [Figure: \(x = y^2 - 2\), \(y = 1\), \(y = -1\), \(x = e^y\)]
4 Areas Between Curves · Level 2
Find the area of the shaded region. [Figure: \(x = y^2 - 4y\), \(x = 2y - y^2\), intersecting at \((-3, 3)\)]
5 Areas Between Curves · Level 3
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \(x\) or \(y\). Draw a typical approximating rectangle and label its height and width. Then find the area of the region. \(y = e^x\), \(y = x^2 - 1\), \(x = -1\), \(x = 1\)
6 Areas Between Curves · Level 3
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \(x\) or \(y\). Draw a typical approximating rectangle and label its height and width. Then find the area of the region. \(y = \sin x\), \(y = x\), \(x = \dfrac{\pi}{2}\), \(x = \pi\)
7 Areas Between Curves · Level 3
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \(x\) or \(y\). Draw a typical approximating rectangle and label its height and width. Then find the area of the region. \(y = (x - 2)^2\), \(y = x\)
8 Areas Between Curves · Level 3
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \(x\) or \(y\). Draw a typical approximating rectangle and label its height and width. Then find the area of the region. \(y = x^2 - 4x\), \(y = 2x\)
9 Areas Between Curves · Level 3
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \(x\) or \(y\). Draw a typical approximating rectangle and label its height and width. Then find the area of the region. \(y = \dfrac{1}{x}\), \(y = \dfrac{1}{x^2}\), \(x = 2\)
10 Areas Between Curves · Level 3
\(y = \sin x\), \(y = \dfrac{2x}{\pi}\), \(x \geq 0\)
11 Areas Between Curves · Level 3
\(x = 1 - y^2\), \(x = y^2 - 1\)
12 Areas Between Curves · Level 3
\(4x + y^2 = 12\), \(x = y\)
13 Areas Between Curves · Level 2
\(y = 12 - x^2\), \(y = x^2 - 6\)
14 Areas Between Curves · Level 2
\(y = x^2\), \(y = 4x - x^2\)
15 Areas Between Curves · Level 3
\(y = \sec^2 x\), \(y = 8 \cos x\), \(-\dfrac{\pi}{3} \leq x \leq \dfrac{\pi}{3}\)
16 Areas Between Curves · Level 3
\(y = \cos x\), \(y = 2 - \cos x\), \(0 \leq x \leq 2 \pi\)
17 Areas Between Curves · Level 3
\(x = 2y^2\), \(x = 4 + y^2\)
18 Areas Between Curves · Level 3
\(y = \sqrt{x - 1}\), \(x - y = 1\)
19 Areas Between Curves · Level 3
\(y = \cos \pi x\), \(y = 4x^2 - 1\)
20 Areas Between Curves · Level 3
\(x = y^4\), \(y = \sqrt{2 - x}\), \(y = 0\)
21 Areas Between Curves · Level 3
\(y = \tan x\), \(y = 2 \sin x\), \(-\dfrac{\pi}{3} \leq x \leq \dfrac{\pi}{3}\)
22 Areas Between Curves · Level 3
\(y = x^3\), \(y = x\)
23 Areas Between Curves · Level 2
\(y = \sqrt{2x}\), \(y = \dfrac{1}{8} x^2\), \(0 \leq x \leq 6\)
24 Areas Between Curves · Level 3
\(y = \cos x\), \(y = 1 - \cos x\), \(0 \leq x \leq \pi\)
25 Areas Between Curves · Level 3
\(y = x^4\), \(y = 2 - |x|\)
26 Areas Between Curves · Level 3
\(y = \sinh x\), \(y = e^{-x}\), \(x = 0\), \(x = 2\)
27 Areas Between Curves · Level 3
\(y = \dfrac{1}{x}\), \(y = x\), \(y = \dfrac{1}{4} x\), \(x > 0\)
28 Areas Between Curves · Level 3
\(y = \dfrac{1}{4} x^2\), \(y = 2x^2\), \(x + y = 3\), \(x \geq 0\)
29 Areas Between Curves · Level 3
The graphs of two functions are shown with the areas of the regions between the curves indicated.
(a) What is the total area between the curves for \(0 \leq x \leq 5\)?
(b) What is the value of \(\displaystyle\int_{0}^{5} [f(x) - g(x)] d x\)? [Figure shows regions with areas 12 and 27]

Enter your answer directly below each part above.

30 Areas Between Curves · Level 3
\(y = \dfrac{x}{\sqrt{1 + x^2}}\), \(y = \dfrac{x}{\sqrt{9 - x^2}}\), \(x \geq 0\)
31 Areas Between Curves · Level 3
\(y = \dfrac{x}{1 + x^2}\), \(y = \dfrac{x^2}{1 + x^3}\)
32 Areas Between Curves · Level 3
\(y = \dfrac{\ln x}{x}\), \(y = \dfrac{(\ln x)^2}{x}\)
33 Areas Between Curves · Level 3
Use calculus to find the area of the triangle with the given vertices. \((0, 0)\), \((3, 1)\), \((1, 2)\)
34 Areas Between Curves · Level 3
Use calculus to find the area of the triangle with the given vertices. \((2, 0)\), \((0, 2)\), \((-1, 1)\)
35 Areas Between Curves · Level 3
Evaluate the integral and interpret it as the area of a region. Sketch the region. \(\displaystyle\int_{0}^{\dfrac{\pi}{2}) |\sin x - \cos 2x| d x\)
36 Areas Between Curves · Level 3
Evaluate the integral and interpret it as the area of a region. Sketch the region. \(\displaystyle\int_{-1}^1 |3^x - 2^x| d x\)
37 Areas Between Curves · Level 4
Use a graph to find approximate \(x\)-coordinates of the points of intersection of the given curves. Then find (approximately) the area bounded by the curves. \(y = x \sin(x^2)\), \(y = x^4\), \(x \geq 0\)
38 Areas Between Curves · Level 4
Use a graph to find approximate \(x\)-coordinates of the points of intersection of the given curves. Then find (approximately) the area bounded by the curves. \(y = \dfrac{x}{(x^2 + 1)^2}\), \(y = x^5 - x\), \(x \geq 0\)
39 Areas Between Curves · Level 4
Use a graph to find approximate \(x\)-coordinates of the points of intersection of the given curves. Then find (approximately) the area bounded by the curves. \(y = 3x^2 - 2x\), \(y = x^3 - 3x + 4\)
40 Areas Between Curves · Level 4
Use a graph to find approximate \(x\)-coordinates of the points of intersection of the given curves. Then find (approximately) the area bounded by the curves. \(y = 1.3^x\), \(y = 2 \sqrt{x}\)
41 Areas Between Curves · Level 4
Graph the region between the curves and use your calculator to compute the area correct to five decimal places. \(y = \dfrac{2}{1 + x^4}\), \(y = x^2\)
42 Areas Between Curves · Level 4
Graph the region between the curves and use your calculator to compute the area correct to five decimal places. \(y = e^{1 - x^2}\), \(y = x^4\)
43 Areas Between Curves · Level 4
Graph the region between the curves and use your calculator to compute the area correct to five decimal places. \(y = \tan^2 x\), \(y = \sqrt{x}\)
44 Areas Between Curves · Level 4
Graph the region between the curves and use your calculator to compute the area correct to five decimal places. \(y = \cos x\), \(y = x + 2 \sin^4 x\)
45 Areas Between Curves · Level 4
Use a computer algebra system to find the exact area enclosed by the curves \(y = x^5 - 6x^3 + 4x\) and \(y = x\).
46 Areas Between Curves · Level 4
Sketch the region in the \(x y\)-plane defined by the inequalities \(x - 2y^2 \geq 0\), \(1 - x - |y| \geq 0\) and find its area.
47 Areas Between Curves - Midpoint Rule · Level 3
Racing cars driven by Chris and Kelly are side by side at the start of a race. The table shows the velocities of each car (in miles per hour) during the first ten seconds of the race. Use the Midpoint Rule to estimate how much farther Kelly travels than Chris does during the first ten seconds.
\(t\) \(v_C\) \(v_K\) \(t\) \(v_C\) \(v_K\)
0 0 0 6 69 80
1 20 22 7 75 86
2 32 37 8 81 93
3 46 52 9 86 98
4 54 61 10 90 102
5 62 71
48 Areas Between Curves - Midpoint Rule · Level 3
The widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in the figure. Use the Midpoint Rule to estimate the area of the pool. [Widths: 6.2, 7.2, 6.8, 5.6, 5.0, 4.8, 4.8]
49 Areas Between Curves - Midpoint Rule · Level 3
A cross-section of an airplane wing is shown. Measurements of the thickness of the wing, in centimeters, at 20-centimeter intervals are 5.8, 20.3, 26.7, 29.0, 27.6, 27.3, 23.8, 20.5, 15.1, 8.7, and 2.8. Use the Midpoint Rule to estimate the area of the wing's cross-section.
50 Areas Between Curves · Level 4
If the birth rate of a population is \(b(t) = 2200 e^{0.024 t}\) people per year and the death rate is \(d(t) = 1460 e^{0.018 t}\) people per year, find the area between these curves for \(0 \leq t \leq 10\). What does this area represent?
51 Areas Between Curves · Level 4
In Example 5, we modeled a measles pathogenesis curve by a function \(f\). A patient infected with the measles virus who has some immunity to the virus has a pathogenesis curve that can be modeled by, for instance, \(g(t) = 0.9 f(t)\).
(a) If the same threshold concentration of the virus is required for infectiousness to begin as in Example 5, on what day does this occur?
(b) Let \(P_3\) be the point on the graph of \(g\) where infectiousness begins. It has been shown that infectiousness ends at a point \(P_4\) on the graph of \(g\) where the line through \(P_3\), \(P_4\) has the same slope as the line through \(P_1\), \(P_2\) in Example 5(b). On what day does infectiousness end?
(c) Compute the level of infectiousness for this patient.

Enter your answer directly below each part above.

52 Areas Between Curves · Level 4
The rates at which rain fell, in inches per hour, in two different locations \(t\) hours after the start of a storm are given by \(f(t) = 0.73 t^3 - 2 t^2 + t + 0.6\) and \(g(t) = 0.17 t^2 - 0.5 t + 1.1\). Compute the area between the graphs for \(0 \leq t \leq 2\) and interpret your result in this context.
53 Areas Between Curves · Level 3
Two cars, A and B, start side by side and accelerate from rest. The figure shows the graphs of their velocity functions.
(a) Which car is ahead after one minute? Explain.
(b) What is the meaning of the area of the shaded region?
(c) Which car is ahead after two minutes? Explain.
(d) Estimate the time at which the cars are again side by side.

Enter your answer directly below each part above.

54 Areas Between Curves · Level 4
The figure shows graphs of the marginal revenue function \(R'\) and the marginal cost function \(C'\) for a manufacturer. [Recall from Section 4.7 that \(R(x)\) and \(C(x)\) represent the revenue and cost when \(x\) units are manufactured.] Assume that \(R\) and \(C\) are measured in thousands of dollars. What is the meaning of the area of the shaded region? Use the Midpoint Rule to estimate the value of this quantity.
55 Areas Between Curves · Level 4
The curve with equation \(y^2 = x^2(x + 3)\) is called Tschirnhausen's cubic. If you graph this curve you will see that part of the curve forms a loop. Find the area enclosed by the loop.
56 Areas Between Curves · Level 4
Find the area of the region bounded by the parabola \(y = x^2\), the tangent line to this parabola at \((1, 1)\), and the \(x\)-axis.
57 Areas Between Curves · Level 4
Find the number \(b\) such that the line \(y = b\) divides the region bounded by the curves \(y = x^2\) and \(y = 4\) into two regions with equal area.
58 Areas Between Curves · Level 4
(a) Find the number \(a\) such that the line \(x = a\) bisects the area under the curve \(y = \dfrac{1}{x^2}\), \(1 \leq x \leq 4\).
(b) Find the number \(b\) such that the line \(y = b\) bisects the area in part (a).

Enter your answer directly below each part above.

59 Areas Between Curves · Level 5
Find the values of \(c\) such that the area of the region bounded by the parabolas \(y = x^2 - c^2\) and \(y = c^2 - x^2\) is 576.
60 Areas Between Curves · Level 5
Suppose that \(0 < c < \dfrac{\pi}{2}\). For what value of \(c\) is the area of the region enclosed by the curves \(y = \cos x\), \(y = \cos(x - c)\), and \(x = 0\) equal to the area of the region enclosed by the curves \(y = \cos(x - c)\), \(x = \pi\), and \(y = 0\)?
61 Areas Between Curves · Level 5
For what values of \(m\) do the line \(y = m x\) and the curve \(y = \dfrac{x}{x^2 + 1}\) enclose a region? Find the area of the region.
62 Areas Between Curves · Level 2
Find the area of the region bounded above by \(y = e^x\), bounded below by \(y = x\), and bounded on the sides by \(x = 0\) and \(x = 1\).
63 Areas Between Curves · Level 2
Find the area of the region enclosed by the parabolas \(y = x^2\) and \(y = 2x - x^2\).
64 Areas Between Curves · Level 3
Find the approximate area of the region bounded by the curves \(y = \dfrac{x}{\sqrt{x^2 + 1}}\) and \(y = x^4 - x\).
65 Areas Between Curves · Level 3
Figure 8 shows velocity curves for two cars, A and B, that start side by side and move along the same road. What does the area between the curves represent? Use the Midpoint Rule to estimate it.
66 Areas Between Curves · Level 4
The pathogenesis curve in Figure 9 has been modeled by \(f(t) = -t(t - 21)(t + 1)\).
(a) If infectiousness begins on day \(t_1 = 10\) and ends on day \(t_2 = 18\), what are the corresponding concentration levels of infected cells?
(b) The level of infectiousness for an infected person is the area between \(N = f(t)\) and the line through the points \(P_1(t_1, f(t_1))\) and \(P_2(t_2, f(t_2))\). Compute the level of infectiousness for this particular patient.

Enter your answer directly below each part above.

67 Areas Between Curves · Level 3
Find the area of the region bounded by the curves \(y = \sin x\), \(y = \cos x\), \(x = 0\), and \(x = \dfrac{\pi}{2}\).
68 Areas Between Curves · Level 3
Find the area enclosed by the line \(y = x - 1\) and the parabola \(y^2 = 2x + 6\).

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