Question 1

A car's velocity is given by $$v(t) = 3t^2$ + 2 (in m/s). $$Approximate the displacement from $$t = 0 to$$ $$t = 2$$ using 2 subintervals and the left endpoint rule.

Accepted Answer

$$12$$

Question 2

Given $$f'(x) = 4x - 1$$ and $$f(2) = 5$$, find $$f(x).$$

Accepted Answer

Question 3

Evaluate $$\int_{0}^{3} (2x + 1) \, dx$$ using the Fundamental Theorem of Calculus.

Accepted Answer

$$15$$

Question 4

Which property of definite integrals allows you to write $$\int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx$$?

Accepted Answer

Reversal of limits

Question 5

Find the average value of $$f(x) = x^2$ on $$[0,2]$$.

Accepted Answer

$$\frac{4}{3}$$

Question 6

Evaluate $$\int_{0}^{\pi} \sin(x) \, dx$$ using symmetry.

Accepted Answer

$$2$$

Question 7

Evaluate $$\int_{0}^{1} (3x^2$ + 2x) \, dx$$ using substitution.

Accepted Answer

$$\frac{5}{3}$$

Question 8

Given $$v(t) = 4t - 2$$, find the displacement from $$t = 0 to$$ $$t = 3.$$

Accepted Answer

$$15$$

Question 9

Find the area between $$y = x^2$ and y = x$ on $$[0,1]$$.

Accepted Answer

$$\frac{1}{6}$$

Question 10

Find the volume of the solid formed by revolving $$y = x^2$ about the x-axis from x = 0$ to x = 1$ using the disk method.

Accepted Answer

$$\frac{\pi}{5}$$

Question 11

Compare the volumes of solids formed by revolving $$y = x$$ and $$y = x^2$ about the y-axis from x = 0$ to x = 1$. Which is greater?

Accepted Answer

The volume formed by $$y = x is $$greater.

Question 12

Find the volume of the solid formed by revolving $$y = x$$ about the line $$y = 2$$ from $$x = 0 to$$ $$x = 1$$ using the washer method.

Accepted Answer

$$\frac{\pi}{6}$$

Question 13

Find the volume of the solid formed by revolving $$y = x^2$ about the y-axis from x = 0$ to x = 1$ using the shell method.

Accepted Answer

$$\frac{2\pi}{5}$$

Question 14

Evaluate $$\frac{d}{dx} \int_{0}^{x^2$} \cos(t) \, dt.$$

Accepted Answer

$$\cos(x^2$)$$

Question 15

Evaluate $$\int_{0}^{2} \frac{1}{1 + x^2$} \, dx$$ using substitution.

Accepted Answer

$$\arctan(2)$$

Question 16

Find the volume of the solid formed by revolving $$y = \sqrt{x}$$ about the x-axis from $$x = 0 to$$ $$x = 4$$ using the disk method.

Accepted Answer

$$\frac{8\pi}{3}$$

Question 17

Find the area between $$y = \sin(x)$$ and $$y = \cos(x) on$$ $$[0, \frac{\pi}{2}].$$

Accepted Answer

$$2 - \sqrt{2}$$

Question 18

Find the volume of the solid formed by revolving $$y = x$$ about the y-axis from $$x = 0 to$$ $$x = 2$$ using the shell method.

Accepted Answer

$$4\pi$$

Question 19

Find the volume of the solid formed by revolving $$y = x^2$ about the x-axis from x = 0$ to x = 2$ using both the disk and shell methods. What is the volume?

Accepted Answer

$$\frac{16\pi}{5}$$

Integration and Applications of Integration in Calculus

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