1. Copyright © Cengage Learning. All rights reserved. 6 The Integral

2. Copyright © Cengage Learning. All rights reserved. 6.4 The Definite Integral: Algebraic Viewpoint and the Fundamental Theorem of Calculus

3. 3 Example 1 – Finding Cost from Marginal Cost The marginal cost of producing baseball caps at a production level of x caps is 4 – 0.001x dollars per cap. Find the total change of cost if production is increased from 100 to 200 caps. Solution: Method 1: Using an Antiderivative: Let C(x) be the cost function. Because the marginal cost function is the derivative of the cost function, we have C(x) = 4 – 0.001x and so C(x) = ∫ (4 – 0.001x) dx

4. 4 Example 1 – Solution = 4x – 0.001 + K = 4x – 0.0005x 2 + K. Although we do not know what to use for the value of the constant K, we can say: Cost at production level of 100 caps = C(100) = 4(100) – 0.0005(100) 2 + K = $395 + K cont’d K is the constant of integration.

5. 5 Example 1 – Solution Cost at production level of 200 caps = C(200) = 4(200) – 0.0005(200) 2 + K = $780 + K. Therefore, Total change in cost = C(200) – C(100) = ($780 + K) – ($395 + K) = $385. cont’d

6. 6 Example 1 – Solution Notice how the constant of integration simply canceled out! So, we could choose any value for K that we wanted (such as K = 0) and still come out with the correct total change. Put another way, we could use any antiderivative of C(x), such as F(x) = 4x – 0.0005x 2 or F(x) = 4x – 0.0005x 2 + 4 compute F(200) – F(100), and obtain the total change, $385. cont’d F(x) is any antiderivative of C(x) whereas C(x) is the actual cost function.

7. 7 Example 1 – Solution Summarizing this method: To compute the total change of C(x) over the interval [100, 200], use any antiderivative F(x) of C(x), and compute F(200) – F(100). Method 2: Using a Definite Integral: Because the marginal cost C(x) is the rate of change of the total cost function C(x), the total change in C(x) over the interval [100, 200] is given by Total change in cost = Area under the marginal cost function curve cont’d

8. 8 = $385. Putting these two methods together gives us the following surprising result: where F(x) is any antiderivative of C(x). Example 1 – Solution cont’d Figure 20 See Figure 20. Using geometry or Riemann sums

9. 9 The Definite Integral: Algebraic Viewpoint and the Fundamental Theorem of Calculus In Example 1, if we replace C(x) by a general continuous function f (x), we can write where F(x) is any antiderivative of f (x). This result is known as the Fundamental Theorem of Calculus.

10. 10 The Definite Integral: Algebraic Viewpoint and the Fundamental Theorem of Calculus The Fundamental Theorem of Calculus (FTC) Let f be a continuous function defined on the interval [a, b] and let F be any antiderivative of f defined on [a, b]. Then Moreover, an antiderivative of f is guaranteed to exist. In Words: Every continuous function has an antiderivative. To compute the definite integral of f (x) over [a, b], first find an antiderivative F(x), then evaluate it at x = b, evaluate it at x = a, and subtract the two answers.

11. 11 The Definite Integral: Algebraic Approach and the Fundamental Theorem of Calculus Quick Example Because F(x) = x 2 is an antiderivative of f (x) = 2x, = F(1) – F(0) = 1 2 – 0 2 = 1.

12. 12 Example 2 – Using the FTC to Calculate a Definite Integral Calculate Solution: To use the FTC, we need to find an antiderivative of 1 – x 2. But we know that We need only one antiderivative, so let’s take F(x) = x – x 3 /3. The FTC tells us that.

13. 13 Applications

14. 14 Example 5 – Oil Spill Your deep ocean oil rig has suffered a catastrophic failure, and oil is leaking from the ocean floor wellhead at a rate of v(t) = 0.08t 2 – 4t + 60 thousand barrels per day (0  t  20), where t is time in days since the failure. Compute the total volume of oil released during the first 20 days. Solution: We calculate Total volume =

15. 15 Example 5 – Solution cont’d