Q: Which of the following is the correct formula for the area under the curve $$f(x) = x^2$ from x = 0$ to x = 2$ using the definition of the definite integral?

$$\displaystyle \$$int_0$$^2 x^2$\,dx = \lim_{n \to \infty} \sum_{i=1}^n \left(\frac{2i}{n}\$$right)^2$$ \cdot \frac{2}{n}$$

Question 1

Which of the following is the correct formula for the area under the curve $$f(x) = x^2$ from x = 0$ to x = 2$ using the definition of the definite integral?

Accepted Answer

$$\displaystyle \$$int_0$$^2 x^2$\,dx = \lim_{n 	o \infty} \sum_{i=1}^n \left(\frac{2i}{n}\$$right)^2$$ \cdot \frac{2}{n}$$

Question 2

Suppose $$f(x) is $$continuous and non-negative on $$[a, b]. $$Which property of definite integrals guarantees that $$\int_a^b f(x)\,dx \geq 0$$?

Accepted Answer

If $$f(x) \geq 0$$ for all $$x in$$ $$[a, b]$$, then $$\int_a^b f(x)\,dx \geq 0.$$

Question 3

A pharmaceutical company wants to find the average concentration of a drug in the bloodstream over 8 hours, given by $$C(t) = 5e$$^{-0.2t}$$ mg/L. What is the average value of C(t$$)$$ on $$[0,8]$$?

Accepted Answer

$$\displaystyle \frac{1}{8} \$$int_0$$^8 5e$$^{-0.2t}$$\,dt = \frac{25}{8} \left[ -e$$^{-0.2t}$$ \right]_0^8$

Question 4

Which of the following integrals is improper?

Accepted Answer

$$\displaystyle \$$int_1$$^\infty \frac{1}{x^2$}\,dx$$

Question 5

Evaluate the following limit as a definite integral: $$\lim_{n 	o \infty} \sum_{i=1}^n \frac{4}{n} \left(1 + \frac{3i}{n}\$$right)^2$$

Accepted Answer

$$\displaystyle \$$int_1$$^4 x^2$\,dx$$ where $$x = 1 + \frac{3i}{n}$$

Question 6

A solid has a base in the $$xy-$$plane bounded by $$y = x^2$ and y = 4$. If cross-sections perpendicular to the x$-axis are squares, what is the volume of the solid?

Accepted Answer

$$\displaystyle \int_{-2}^2 (4 - x^2$)^2$$\,dx$

Question 7

Which of the following is the correct partial fraction decomposition for $$\displaystyle \frac{3x+4}{x^2$ + x - 6}$$?

Accepted Answer

$$\displaystyle \frac{A}{x-2} + \frac{B}{x+3}$$

Question 8

Which trigonometric substitution is appropriate for evaluating $$\displaystyle \int \frac{dx}{\sqrt{9 - x^2$}}$$?

Accepted Answer

Let $$x = 3 \sin 	heta$$

Question 9

Which of the following series converges?

Accepted Answer

$$\displaystyle \sum_{n=1}^\infty \frac{1}{n^2$}$$

Question 10

Let $$f(x) = \cosh x. $$Which of the following is the correct derivative?

Accepted Answer

$$\displaystyle \frac{d}{dx} \cosh x = \sinh x$$

Question 11

A region bounded by $$y = x^2$ and y = 4$ is revolved about the x$-axis. Which method and integral correctly computes the volume?

Accepted Answer

Washer method: $$\displaystyle \pi \int_{-2}^2 (4^2$ - (x^2$)^2$$)\,dx$

Question 12

Which of the following is the Maclaurin series for $$\sin x$$?

Accepted Answer

$$\displaystyle \sum_{n=0}^\infty \frac{(-1)^n x$$^{2n+1}$$}{(2n+1)!}$$

Question 13

Which of the following statements about the convergence of the geometric series $$\sum_{n=0}^\infty ar^n is $$correct?

Accepted Answer

It converges for $$|r| \u003c 1$$ and diverges otherwise.

Question 14

Which of the following is the correct formula for the inverse hyperbolic sine function in terms of natural logarithms?

Accepted Answer

$$\displaystyle \$$sinh^{-1}$$ x = \ln\left(x + \sqrt{x^2$ + 1}\right)$$

Question 15

A tank is being filled with water at a rate given by $$R(t) = 10 - t$$ liters per minute for $$0 \leq t \leq 10. $$What is the total volume of water added to the tank in the first 10 minutes?

Accepted Answer

$$\displaystyle \$$int_0$$^{10} (10 - t)\,dt$$

Question 16

Which of the following is the correct reduction formula for $$\int \cos^n x\,dx$$?

Accepted Answer

$$\displaystyle \int \cos^n x\,dx = \frac{\cos$$^{n-1}$$ x \sin x}{n} + \frac{n-1}{n} \int \cos$$^{n-2}$$ x\,dx$$

Question 17

Which of the following is the correct statement of the Fundamental Theorem of Calculus, Part 2 (FTC2)?

Accepted Answer

$$\displaystyle \int_a^b f(x)\,dx = F(b) - F(a)$$, where $$F is $$any antiderivative of $$f$$

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