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Which of the following represents the derivative of h(x) = (4x2 – 3x + 5)(2x – 7)? A. h'(x) = 8x2 + 10x + 4 B. h'(x) = 24x2 – 68x + 31 C. h'(x) = 8x3 – 34x2 + 57x – 41 D. h'(x) = 32x3 – 36x2 + 53x – 29 5–Minute Check 4

Which of the following represents the derivative of h(x) = (4x2 – 3x + 5)(2x – 7)? A. h'(x) = 8x2 + 10x + 4 B. h'(x) = 24x2 – 68x + 31 C. h'(x) = 8x3 – 34x2 + 57x – 41 D. h'(x) = 32x3 – 36x2 + 53x – 29 5–Minute Check 4

Approximate the area under a curve using rectangles. You computed limits algebraically using the properties of limits. (Lesson 12-2) Approximate the area under a curve using rectangles. Approximate the area under a curve using definite integrals and integration. Then/Now

regular partition definite integral lower limit upper limit right Riemann sum integration Vocabulary

Area Under a Curve Using Rectangles Approximate the area between the curve f(x) = –x2 + 18x and the x-axis on the interval [0, 18] using 6, 9, and 18 rectangles. Use the right endpoint of each rectangle to determine the height. Example 1

Area using 6 rectangles total area = 945 Area Under a Curve Using Rectangles Area using 6 rectangles total area = 945 Example 1

Area using 9 rectangles total area = 960 Area Under a Curve Using Rectangles Area using 9 rectangles total area = 960 Example 1

Area Under a Curve Using Rectangles Area using 18 rectangles Example 1

Area Under a Curve Using Rectangles total area = 969 Example 1

A. 2 rectangles = 125 units2; 5 rectangles = 160 units2 Approximate the area between the curve f(x) = –x2 + 10x and the x-axis on the interval [0, 10] using 2 and 5 rectangles. Use the right endpoint of each rectangle to determine the height. A. 2 rectangles = 125 units2; 5 rectangles = 160 units2 B. 2 rectangles = 125 units2; 5 rectangles = 128 units2 C. 2 rectangles = 25 units2; 5 rectangles = 128 units2 D. 2 rectangles = 16 units2; 5 rectangles = 160 units2 Example 1

Area Under a Curve Using Left and Right Endpoints Approximate the area between the curve f(x) = x2 + 1 and the x-axis on the interval [0, 4] by first using the right endpoints and then by using the left endpoints of the rectangles. Use rectangles with a width of 1. Then find the average for both approximations. Use the figures for reference. Using right endpoints for the height of each rectangle produces four rectangles with a width of 1 unit. Using left endpoints for the height of each rectangle produces four rectangles with a width of 1 unit. Example 2

Area using right endpoints Area Under a Curve Using Left and Right Endpoints Area using right endpoints total area = 34 units2 Example 2

Area using left endpoints Area Under a Curve Using Left and Right Endpoints Area using left endpoints total area = 18 units2 Example 2

Area Under a Curve Using Left and Right Endpoints Answer: The area using the right and left endpoints is 34 and 18 square units, respectively. We now have lower and upper estimates for the area of the region, 18 < area < 34. Averaging the two areas would give a better approximation of 26 square units. Example 2

Approximate the area between the curve f(x) = 2x2 + 2 and the x-axis on the interval [0, 3] by first using the right endpoints and then by using the left endpoints of the rectangles. Use rectangles with a width of 1. Then find the average for both approximations. A. right endpoint = 16 units2; left endpoint = 68 units2; average = 42 units2 B. right endpoint = 68 units2; left endpoint = 16 units2; average = 42 units2 C. right endpoint = 16 units2 left endpoint = 34 units2; average = 25 units2 D. right endpoint = 34 units2 left endpoint = 16 units2; average = 25 units2 Example 2

Key Concept 3

Area Under a Curve Using Integration Use limits to find the area of the region between the graph of y = x2 + 1 and the x-axis on the interval [0, 4], or Example 3

First, find x and xi. Formula for x b = 4 and a = 0 Formula for xi Area Under a Curve Using Integration First, find x and xi. Formula for x b = 4 and a = 0 Formula for xi a = 0 and Example 3

Calculate the definite integral that gives the area. Area Under a Curve Using Integration Calculate the definite integral that gives the area. Definition of a definite integral f(xi) = (xi)2 + 1 Example 3

Factor. Simplify. Apply summations. Area Under a Curve Using Integration Factor. Simplify. Apply summations. Example 3

Factor constants. Summation formulas Distribute Area Under a Curve Using Integration Factor constants. Summation formulas Distribute Example 3

Factor and perform division. Area Under a Curve Using Integration Simplify. Factor and perform division. Limit Theorems Example 3

Limits Simplify. Answer: Area Under a Curve Using Integration Example 3

Use limits to find the area of the region between the graph of y = 2x2 and the x-axis on the interval [0, 3], or A. 9 units2 B. 18 units2 C. 27 units2 D. about 18.67 units2 Example 3

Area Under a Curve Using Integration Use limits to find the area of the region between the graph of y = x3 + 1 and the x-axis on the interval [2, 4], or Example 4

First, find x and xi. Formula for x b = 4 and a = 2 Formula for xi Area Under a Curve Using Integration First, find x and xi. Formula for x b = 4 and a = 2 Formula for xi a = 2 and Example 4

Calculate the definite integral that gives the area. Area Under a Curve Using Integration Calculate the definite integral that gives the area. Definition of a definite integral f(xi) = (xi)3 + 1 Example 4

Factor. Expand. Simplify. Area Under a Curve Using Integration Example 4

Apply summations. Factor constants. Area Under a Curve Using Integration Apply summations. Factor constants. Example 4

Area Under a Curve Using Integration Summation formulas Example 4

Area Under a Curve Using Integration Distribute Example 4

Factor and perform division. Area Under a Curve Using Integration Simplify. Factor and perform division. Example 4

Limit theorems. Simplify. Answer: 62 units2 Area Under a Curve Using Integration Limit theorems. Simplify. Answer: 62 units2 Example 4

Use limits to approximate the area of the region between the graph of y = 2x3 + 3 and the x-axis on the interval [1, 3], or A. 50 units2 B. 64 units2 C. 46 units2 D. 18 units2 Example 4

Area Under a Curve BUSINESS A clothing manufacturer produces 2000 pairs of pants per day. The cost for increasing the number of pairs per day from 2000 to 5000 can be found by . What is the increase in cost? Example 5

First, find x and xi. Formula for x b = 5000 and a = 2000 Area Under a Curve First, find x and xi. Formula for x b = 5000 and a = 2000 Formula for xi a = 2000 and Example 5

Calculate the definite integral that gives the area. Area Under a Curve Calculate the definite integral that gives the area. Definition of a definite integral f(xi) = 20 – 0.004xi Example 5

Distributive property and simplify. Area Under a Curve Distributive property and simplify. Example 5

Apply summations. Factor. Summation formulas. Area Under a Curve Example 5

Factor and perform division. Area Under a Curve Distribute Simplify. Factor and perform division. Example 5

Limit Theorems Simplify. Area Under a Curve Limit Theorems Simplify. Answer: The cost for increasing the number of pairs of pants produced per day from 2000 to 5000 pairs is $18,000. Example 5

PAVING Millie is putting in a brick patio PAVING Millie is putting in a brick patio. The paver charges $105 per square foot. If the area of Millie’s patio can be found by , to the nearest dollar, how much will the paver charge for installing the patio if x is given in feet? A. $2,017.50 B. $3150 C. $3500 D. $4,120.10 Example 5

regular partition definite integral lower limit upper limit right Riemann sum integration Vocabulary