Question 1

Evaluate the integral $$\int (3x$$^{1/3}$$ - 4x$$^{-1/3}$$ + 6) \, dx.$$

Accepted Answer

$$\frac{9}{4}x$$^{4/3}$$ - 6x$$^{2/3}$$ + 6x + C$$

Question 2

A ball is thrown vertically upward with acceleration $$a(t) = -32. If $$its velocity at $$t=1 is$$ $$20 ft/s $$and its position at $$t=2 is$$ $$0$$, what is its position function $$s(t)$$?

Accepted Answer

$$s(t) = -16t^2$ + 52t - 40$$

Question 3

Evaluate the definite integral $$\int_{-1}^{2} (2x - 12x^3$) \, dx.$$

Accepted Answer

$$7/2$$

Question 4

Which of the following is the correct antiderivative of $$f(x) = 2x(x^2$ - 1$$)^{99}$$?

Accepted Answer

$$\frac{(x^2$ - 1$$)^{100}$$}{100} + C$$

Question 5

A car's acceleration is given by $$a(t) = 2e^t - 12. If$$ $$v(0) = 1$$ and $$s(0) = 3$$, what is the position function $$s(t)$$?

Accepted Answer

$$s(t) = 2e^t - 6t^2$ - t + 1$$

Question 6

Evaluate the integral $$\int 4x \sqrt{x^2$ + 8} \, dx.$$

Accepted Answer

$$\frac{4}{3}(x^2$ + 8$$)^{3/2}$$ + C$$

Question 7

A pharmaceutical company models the concentration of a drug in the bloodstream by $$f(x) = x^2$ + 1$$ over the interval $$[5, 10]. $$Using a left Riemann sum with 10 subintervals, what is the approximate area under the curve?

Accepted Answer

$$278.125$$

Question 8

Evaluate the integral $$\int x e$$^{x^2}$$ \, dx.$$

Accepted Answer

$$\frac{1}{2}e$$^{x^2}$$ + C$$

Question 9

Find the net area of the region bounded by $$f(x) = x^2$ - 25$$ and the $$x-$$axis on $$[2, 4].$$

Accepted Answer

$$-\frac{94}{3}$$

Question 10

A factory's production rate is modeled by $$f(x) = 8 - 2x on$$ $$[0, 6]. $$What is the total area under the curve (regardless of sign) and the net area (taking sign into account)?

Accepted Answer

Area: $$20$$, Net Area: $$12$$

Question 11

Evaluate the integral $$\int \sin$$^{10}$$(x) \cos(x) \, dx.$$

Accepted Answer

$$\frac{1}{11}\sin$$^{11}$$(x) + C$$

Question 12

A biologist is studying a population modeled by $$f(x) = x(x+1)(x-1) on$$ $$[-1, 2]. $$What is the net area between the curve and the $$x-$$axis on this interval?

Accepted Answer

$$\frac{9}{4}$$

Question 13

Evaluate the integral $$\int 5x(12x^3$ - 10x) \, dx.$$

Accepted Answer

$$12x^5$$ - \frac{50}{3}$$x^3 + C$

Question 14

Evaluate the definite integral $$\$$int_0$$^1 2x(4 - x^2$) \, dx.$$

Accepted Answer

$$-42$$

Question 15

A chemical reaction's rate is given by $$f(x) = x^2$ + 1 on$$ $$[5, 10]. $$Using a midpoint Riemann sum with 10 subintervals, what is the approximate area under the curve?

Accepted Answer

$$296.5625$$

Question 16

Evaluate the integral $$\int 2x^2$ \sqrt{1 - 4x^3$} \, dx.$$

Accepted Answer

$$-\frac{1}{3}\sqrt{1 - 4x^3$} + C$$

Question 17

Evaluate the integral $$\int_{3}^{5} \frac{2x}{(x^2$ + 1)^2$} \, dx.$$

Accepted Answer

$$\frac{4}{65}$$

Question 18

A company wants to estimate the area under the curve $$f(x) on$$ $$[0,6]$$ using 3 subintervals. If the left, right, and midpoint Riemann sums are $$32$$, $$52$$, and $$42$$ respectively, which of the following is correct?

Accepted Answer

Left: $$32$$, Right: $$52$$, Midpoint: $$42$$

Question 19

Evaluate the integral $$\int (\sqrt{x} + 1)^4$ 2\sqrt{x} \, dx.$$

Accepted Answer

$$\frac{1}{5}(\sqrt{x} + 1)^5$ + C$$

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