MATH 2B

Midterm 1 | Winter '17 | Youssefpour

1. Express the following limit as a definite integral on the interval $[0, 1]$ $\lim_{n \to x} \sum_{i = 1}^{n} \frac{e^{2x_i}}{1 + \sin x_i} \delta x$ You do not need to evaluate the integral.

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2. Evalute the following indefinite integrals.
a) $\int x^3 \sqrt[3]{x^2 + 1} dx$

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$\int \frac{\sin(\ln t)}{t} dt$

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3. Water leaks out from the bottom of a storage tank at a rate of $r(t) = 100 - 2t$ liters per minutes for $0 \le t \le 50$. Find the amount of water that leaks out during the first $20$ minutes?

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4. Evalute the following definite integrals if they exist Show all your work)
a) $\int_0^1 \frac{e^x}{1 + e^{2x}} dx$

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b) $\int_0^1 (\sqrt[5]{5} + 1) dt$

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5. True or False (you do not need to provide any explanation)
a) If $f$ is continuous on $[a, b]$, then $\frac{d}{dx}(\int_a^b f(x) dx) = f(x)$

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b) If $f$ is continuous on $[a, b]$, then $\int_a^b f(x) dx = -2 \int_b^a f(x) dx$

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c) If $f$ is continuous on $[a, b]$, then $\int_a^b 4f(x) dx = 4\int_a^b f(x) dx$

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d) If $f$ is continuous on $[a, b]$, then $\int_a^b \sqrt{f(x)} dx = \sqrt{\int_a^b f(x) dx}$

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6. Find the volume of the solid obtained by rotating the region enclosed by the curves, $y = x^2$ and $y = x$ about the line $x = -2$

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7. Find the area of the region enclosed by the curves $y = x^2 - 4x$ and $y = 2x$.

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8. If $f(x) = \int_3^{x^4} \sqrt{t^2 + 6} dt$, find $f'(x)$.

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9. If $g(t)$ is a continuous function on the interval $[0, \pi]$, $\int_0^{\pi/4} g(t) dt = \frac{\pi}{4}, \int_{\pi/4}^{\pi/2} g(t) dt = \frac{\pi}{4}$ and $\int_{\pi/2}^{\pi} g(t) dt = \frac{\pi}{2}$, then evaluate $\int_0^{\pi} (g(t) - \cos t) dt$

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