The definite integral

Suppose that the following information is known about the functions $f$, $g$, $x^2$, and $x^3$:

• $\int_0^2 f(x) \, dx = -3$; $\int_2^5 f(x) \, dx = 2$

• $\int_0^2 g(x) \, dx = 4$; $\int_2^5 g(x) \, dx = -1$

• $\int_0^2 x^2 \, dx = \frac{8}{3}$; $\int_2^5 x^2 \, dx = \frac{117}{3}$

• $\int_0^2 x^3 \, dx = 4$; $\int_2^5 x^3 \, dx = \frac{609}{4}$

Use the provided information and the rules discussed in the preceding section to evaluate each of the following definite integrals.

1. $\int_5^2 f(x) \, dx$

2. $\int_0^5 g(x) \, dx$

3. $\int_0^5 (f(x) + g(x))\, dx$

4. $\int_2^5 (3x^2 - 4x^3) \, dx$

5. $\int_5^0 (2x^3 - 7g(x)) \, dx$

Suppose that $v(t) = \sqrt{4-(t-2)^2}$ tells us the instantaneous velocity of a moving object on the interval $0 \le t \le 4$, where $t$ is measured in minutes and $v$ is measured in meters per minute.

1. Sketch an accurate graph of $y = v(t)$. What kind of curve is $y = \sqrt{4-(t-2)^2}$?

2. Evaluate $\int_0^4 v(t) \, dt$ exactly.

3. In terms of the physical problem of the moving object with velocity $v(t)$, what is the meaning of $\int_0^4 v(t) \, dt$? Include units on your answer.

4. Determine the exact average value of $v(t)$ on…