Q: Find $$\frac{dy}{dx} if$$ $$y = 3 \tanh 7x^2$.

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

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 plate in the shape of a right triangle with legs 6 ft and 8 ft is submerged in a liquid of weight density $$31.2 lb/ft$^3$$ with the 6 ft side at the surface. What is the fluid force on the plate?

Accepted Answer

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

Question 5

A semicircular plate of radius 3 in is submerged vertically in a liquid with the flat side at the surface. What is the fluid force on the plate if the weight density is $$62.4 lb/ft$^3$$? (Express your answer as an integral.)

Accepted Answer

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

Question 6

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

$$(\bar{x}, \bar{y}) = \left(\frac{\$$int_1$$^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 7

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

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

Question 8

Evaluate $$12 \cosh(\ln 4). ($$Recall $$\cosh(x) = \frac{e^x + e$$^{-x}$$}{2}.)$$

Accepted Answer

$$\frac{12}{2}(4 + \frac{1}{4}) = 25.5$$

Question 9

Evaluate $$\$$sinh^{-1}$$ 3. ($$Express as a logarithm.)

Accepted Answer

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

Question 10

Find $$\frac{dy}{dx} if$$ $$y = 3 	anh 7x^2$.

Accepted Answer

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

Question 11

Find $$\frac{dy}{dx} if$$ $$y = -8 \$$cosh^{-1}$$ 3x.$$

Accepted Answer

$$\frac{-8}{\sqrt{9x^2$ - 1}}$$

Question 12

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

Accepted Answer

$$\$$int_0$$^{\ln 2} \cosh x \, dx = \sinh x \Big|_0$$^{\ln 2}$$

Question 13

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

Accepted Answer

$$\frac{1}{3} 	anh(3e^t)$$

Question 14

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

Accepted Answer

$$\frac{1}{12} \arcsin(4x) + C$$

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