Stewart Precalc 6e Section 13.5: Areas

1 Concept - Approximating Area · Level 1

The graph of a function \(f\) is shown. To find the area under the graph of \(f\), we first approximate the area by ___. Approximate the area by drawing four rectangles. The area \(R_4\) of this approximation is ___

2 Concept - Exact Area · Level 1

Let \(R_n\) be the approximation obtained by using \(n\) rectangles of equal width. The exact area under the graph of \(f\) is ___

3 Skills - Lower and Upper Estimates · Level 2

(a) By reading values from the given graph of \(f\), use five rectangles to find a lower estimate and an upper estimate for the area under the given graph of \(f\) from \(x = 0\) to \(x = 10\). In each case, sketch the rectangles that you use.

Answer for 3(a)

(b) Find new estimates using ten rectangles in each case.

Answer for 3(b)

Enter your answer directly below each part above.

4 Skills - Left and Right Endpoint Approximations · Level 2

(a) Use six rectangles to find estimates of each type for the area under the given graph of \(f\) from \(x = 0\) to \(x = 12\). (i) \(L_6\) (using left endpoints) (ii) \(R_6\) (using right endpoints)

Answer for 4(a)

(b) Is \(L_6\) an underestimate or an overestimate of the true area?

Answer for 4(b)

(c) Is \(R_6\) an underestimate or an overestimate of the true area?

Answer for 4(c)

Enter your answer directly below each part above.

5 Skills - Approximating Area · Level 2

Use the graph to estimate the area under the graph of \(f(x) = \left(\dfrac{1}{2}\right) x + 2\) over the indicated interval.

6 Skills - Approximating Area · Level 2

Use the graph to estimate the area under the graph of \(f(x) = 4 - x^2\) over the indicated interval.

7 Skills - Approximating Area · Level 2

Use the graph to estimate the area under the graph of \(f(x) = \dfrac{4}{x}\) over the indicated interval.

8 Skills - Approximating Area · Level 2

Use the graph to estimate the area under the graph of \(f(x) = 9 x - x^3\) over the indicated interval.

9 Skills - Rectangle Approximations · Level 2

(a) Estimate the area under the graph of \(f(x) = \dfrac{1}{x}\) from \(x = 1\) to \(x = 5\) using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate?

Answer for 9(a)

(b) Repeat part (a), using left endpoints.

Answer for 9(b)

Enter your answer directly below each part above.

10 Skills - Rectangle Approximations · Level 2

(a) Estimate the area under the graph of \(f(x) = 25 - x^2\) from \(x = 0\) to \(x = 5\) using five approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate?

Answer for 10(a)

(b) Repeat part (a) using left endpoints.

Answer for 10(b)

Enter your answer directly below each part above.

11 Skills - Rectangle Approximations · Level 2

(a) Estimate the area under the graph of \(f(x) = 1 + x^2\) from \(x = -1\) to \(x = 2\) using three rectangles and right endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles.

Answer for 11(a)

(b) Repeat part (a) using left endpoints.

Answer for 11(b)

Enter your answer directly below each part above.

12 Skills - Rectangle Approximations · Level 3

(a) Estimate the area under the graph of \(f(x) = e^{-x}\), \(0 \leq x \leq 4\), using four approximating rectangles and taking the sample points to be (i) right endpoints (ii) left endpoints In each case, sketch the curve and the rectangles.

Answer for 12(a)

(b) Improve your estimates in part (a) by using eight rectangles.

Answer for 12(b)

Enter your answer directly below each part above.

13 Skills - Area as Limit · Level 3

Use the definition of area as a limit to find the area of the region that lies under the curve. Check your answer by sketching the region and using geometry. \(y = 3 x\), \(0 \leq x \leq 5\)

14 Skills - Area as Limit · Level 3

Use the definition of area as a limit to find the area of the region that lies under the curve. Check your answer by sketching the region and using geometry. \(y = 2 x + 1\), \(1 \leq x \leq 3\)

15 Skills - Area Under Curve · Level 3

Find the area of the region that lies under the graph of \(f\) over the given interval. \(f(x) = 3 x^2\), \(0 \leq x \leq 2\)

16 Skills - Area Under Curve · Level 3

Find the area of the region that lies under the graph of \(f\) over the given interval. \(f(x) = x + x^2\), \(0 \leq x \leq 1\)

17 Skills - Area Under Curve · Level 3

Find the area of the region that lies under the graph of \(f\) over the given interval. \(f(x) = x^3 + 2\), \(0 \leq x \leq 5\)

18 Skills - Area Under Curve · Level 3

Find the area of the region that lies under the graph of \(f\) over the given interval. \(f(x) = 4 x^3\), \(2 \leq x \leq 5\)

19 Skills - Area Under Curve · Level 3

Find the area of the region that lies under the graph of \(f\) over the given interval. \(f(x) = x + 6 x^2\), \(1 \leq x \leq 4\)

20 Skills - Area Under Curve · Level 3

Find the area of the region that lies under the graph of \(f\) over the given interval. \(f(x) = 20 - 2 x^2\), \(2 \leq x \leq 3\)

21 Discovery - Calculator Area Program · Level 2

Approximating Area with a Calculator. When we approximate areas using rectangles, the more rectangles we use, the more accurate the answer. The following TI-83 program finds the approximate area under the graph of \(f\) on the interval \([a, b]\) using \(n\) rectangles. To use the program, first store the function \(f\) in \(Y_1\). The program prompts you to enter N, the number of rectangles, and A and B, the endpoints of the interval.

(a) Approximate the area under the graph of \(f(x) = x^5 + 2 x + 3\) on \([1, 3]\), using 10, 20, and 100 rectangles.

Answer for 21(a)

(b) Approximate the area under the graph of \(f\) on the given interval, using 100 rectangles. (i) \(f(x) = \sin x\), on \([0, \pi]\) (ii) \(f(x) = e^{-x^2}\), on \([-1, 1]\) PROGRAM: AREA :Prompt N :Prompt A :Prompt B :(B-A)/N -> D :0 -> S :A -> X :For (K,1,N) :X+D -> X :S+Y_1 -> S :End :D*S -> S :Disp "AREA IS" :Disp S

Answer for 21(b)

Enter your answer directly below each part above.

22 Discovery - Polygons vs Curved Boundaries · Level 1

Regions with Straight Versus Curved Boundaries. Write a short essay that explains how you would find the area of a polygon, that is, a region bounded by straight line segments. Then explain how you would find the area of a region whose boundary is curved, as we did in this section. What is the fundamental difference between these two processes?

23 Example - Estimating Area with Rectangles · Level 3

Use rectangles to estimate the area under the parabola \(y = x^2\) from \(0\) to \(1\) (the parabolic region \(S\) illustrated in Figure 3).

24 Example - The Limit of Approximating Sums · Level 3

For the region \(S\) in Example 1 (under \(y = x^2\) on \([0, 1]\)), show that the sum of the areas of the upper approximating rectangles approaches \(\dfrac{1}{3}\), that is, \(\operatorname*{lim}\limits_{n\rightarrow \infty} R_n = \dfrac{1}{3}\).

25 Example - Finding the Area Under a Curve · Level 3

Find the area of the region that lies under the parabola \(y = x^2\), \(0 \leq x \leq 5\).

26 Example - Finding the Area Under a Curve · Level 3

Find the area of the region that lies under the parabola \(y = 4 x - x^2\), \(1 \leq x \leq 3\).

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