Q: Find the first derivative of $$y = 3 \tanh 7x^2$.

$$3 \cdot \text{sech}^2(7x^2$) \cdot 14x$$

Q: Evaluate $$\int e^t \text{sech}^2(e^t) \tanh(3e^t) \, dt.$$

$$\frac{1}{3} \text{sech}^2(3e^t)$$

Question 1

Find the length of the arc of $$x = 13y^2$ + \frac{23}{2} on $$the interval $$0 \u003c y \u003c 3.$$

Accepted Answer

$$\$$int_0$$^3 \sqrt{1 + (26y)^2$} \, dy$$

Question 2

Find the length of the curve $$y = 18x^4$ + 14x^2$ from x = 1$ to x = 2$.

Accepted Answer

$$\$$int_1$$^2 \sqrt{1 + (72x^3$ + 28x)^2$} \, dx$$

Question 3

Find the area of the surface generated by revolving the curve $$y = 23x$$^{3/2}$$ - 12x$$^{1/2}$$ between (1$$, \frac{1}{6})$$ and (9$$, \frac{33}{2})$$ about the x-axis.

Accepted Answer

$$2\pi \$$int_1$$^9 \$$left(23x^{3/2}$$ - 12x$$^{1/2}$$\right) \sqrt{1 + \left(\frac{d}{dx}(23x$$^{3/2}$$ - 12x$$^{1/2}$$)\$$right)^2$$} \, dx$$

Question 4

A vertical triangular plate (see image 1) is submerged in a liquid with weight density $$31.2 lb/ft$^3. $$What is the fluid force against the plate?

Accepted Answer

$$31.2 \$$int_0$$^8 y \frac{6}{8} (8-y) \, dy$$

Question 5

Find the centroid $$(\bar{x}, \bar{y}) of $$the region bounded by $$y = 4 - x^2$, y = 0$, x = 1$, and x = 2$.

Accepted Answer

$$\left( \frac{\$$int_1$$^2 x(4-x^2$) \, dx}{\$$int_1$$^2 (4-x^2$) \, dx}, \frac{\$$int_1$$^2 \frac{1}{2}(4-x^2$)^2$$ \, dx}{\$$int_1^2$$ $$(4-x^2$$) \, dx} \right)$

Question 6

Find the center of mass of a lamina with density $$\rho = 3x$$ bounded by $$y = 4x^2$(x-1)$$ and the x-axis.

Accepted Answer

$$\left( \frac{\int x \cdot 3x \cdot 4x^2$(x-1) \, dx}{\int 3x \cdot 4x^2$(x-1) \, dx}, \frac{\int 2x^2$(x-1) \cdot 3x \, dx}{\int 3x \cdot 4x^2$(x-1) \, dx} \right)$$

Question 7

Evaluate $$12 \cosh(\ln 4).$$

Accepted Answer

$$12 \cdot \frac{4 + 1/4}{2} = 25.5$$

Question 8

Evaluate $$\$$sinh^{-1}$$ 3.$$

Accepted Answer

$$\ln(3 + \sqrt{3^2$ + 1})$$

Question 9

Find the first derivative of $$y = 3 	anh 7x^2$.

Accepted Answer

$$3 \cdot 	ext{sech}^2(7x^2$) \cdot 14x$$

Question 10

Find the arc length of the catenary $$y = \cosh x$$ between $$x = 0$$ and $$x = \ln 2.$$

Accepted Answer

$$\$$int_0$$^{\ln 2} \sqrt{\$$cosh^2$$ x} \, dx$$

Question 11

Evaluate $$\int e^t 	ext{sech}^2(e^t) 	anh(3e^t) \, dt.$$

Accepted Answer

$$\frac{1}{3} 	ext{sech}^2(3e^t)$$

Question 12

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

Accepted Answer

$$\frac{1}{12} 	ext{arccosh}(2x) + C$$

Question 13

A semicircular plate of radius $$3 in is $$submerged vertically in a liquid (see image 2). If the weight density of the liquid is $$62.4 lb/ft$^3$$, what is the fluid force on the plate?

Accepted Answer

$$62.4 \$$int_0$$^3 2\sqrt{9-y^2$} (3-y) \, dy$$

Question 14

Suppose a lamina occupies the region bounded by $$y = x^2$ and y = 4$ with density $$\rho(x, y) = x$$. What is the x$-coordinate of the center of mass?

Accepted Answer

$$\frac{\int_{-2}^2 x \int_{x^2$}^4 x \, dy \, dx}{\int_{-2}^2 \int_{x^2$}^4 x \, dy \, dx}$$

MAC 2312 Exam 1 Review – Calculus Study Guidance

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