Question 1

Which of the following is the correct integral for finding the area between the curves $$y = 6x$$ and $$y = 3x^2$ - 6x$$ from their points of intersection?

Accepted Answer

$$\int_{0}^{2} (6x - (3x^2$ - 6x)) \, dx$$

Question 2

Evaluate the integral $$\int \frac{3x^2$}{x+4} \, dx$$ using algebraic manipulation.

Accepted Answer

$$\int (3x - 12 + \frac{48}{x+4}) \, dx$$

Question 3

A tank in the shape of a right circular cone is 12 feet across the top and 9 feet high. If the tank is filled to a depth of 3 feet, which integral represents the work required to fill it (assume water weighs $$62.4 lb/ft$^3$$)?

Accepted Answer

$$\int_{0}^{3} 62.4 \pi \left(\frac{4}{3} y \$$right)^2$$ y \, dy$$

Question 4

Find the arc length of the curve $$y = \frac{1}{3}x$$^{\frac{3}$${2}} - 1 on $$the interval $$[2,6].$$

Accepted Answer

$$\int_{2}^{6} \sqrt{1 + \left(\frac{1}{2}x$$^{\frac{1}$${2}}\$$right)^2$$} \, dx$$

Question 5

A ten-meter chain with density $$1.5 kg/m $$hangs from a platform $$11$$ meters above the ground. What is the correct integral to compute the work required to lift the chain to the platform?

Accepted Answer

$$\int_{0}^{10} 1.5 \cdot 9.8 \cdot (11 - y) \, dy$$

Question 6

Which formula represents the surface area generated by revolving $$y = 2\sqrt{x} on$$ $$[1,3]$$ about the $$x-$$axis?

Accepted Answer

$$\int_{1}^{3} 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\$$right)^2$$} \, dx$$

Question 7

Given the lamina of uniform density $$\rho$$ bounded by $$y = \frac{1}{3}x^2$, y = 0$, and x = 2$, which integral represents the moment about the y$-axis (M_y$)?

Accepted Answer

$$\rho \int_{0}^{2} x \left(\frac{1}{3}x^2$ - 0\right) \, dx$$

Question 8

Which of the following is the exponential form of the hyperbolic cosine function $$\cosh x$$?

Accepted Answer

$$\frac{e$$^{x}$$ + e$$^{-x}$$}{2}$$

Question 9

Evaluate $$\cosh 0$$ using the exponential definition.

Accepted Answer

$$1$$

Question 10

Find the center of mass $$(\bar{x}, \bar{y})$$ for the system of point masses: $$m_1 = 8$ at (-3$$, -1)$$, m_2$ = 1 at$$ $$(0, 0)$$, $$m_3 = 4$ at (-1$$, 2)$$.

Accepted Answer

$$\left(\frac{-32}{13}, \frac{6}{13}\right)$$

Question 11

Which integral represents the area of the surface generated by revolving $$y = \sqrt{25 - x^2$} on$$ $$[-4,4]$$ about the $$x-$$axis?

Accepted Answer

$$\int_{-4}^{4} 2\pi \sqrt{25 - x^2$} \sqrt{1 + \left(\frac{-x}{\sqrt{25 - x^2$}}\$$right)^2$$} \, dx$$

Question 12

A spring requires $$6$$ joules of work to stretch $$0.5$$ meters from its natural length. What is the work required to stretch it an additional $$0.25$$ meters?

Accepted Answer

$$\frac{9}{8}$$ joules

Question 13

Which integral represents the volume of the solid generated by revolving the region bounded by $$y = x^3$ and y$$ = \sqrt{x}$$ about the line y = -2$?

Accepted Answer

$$\int_{a}^{b} \pi \left[(\sqrt{x} + 2)^2$ - (x^3$ + 2)^2$\right] \, dx$$

Question 14

Find the derivative $$\frac{d}{dx}(	ext{sech}^3 x).$$

Accepted Answer

$$-3 	ext{sech}^3 x 	anh x$$

Question 15

Which integral represents the volume of the solid generated by revolving the region bounded by $$y = 16x - x^2$, x = 0$, and y = 64$ about the y$-axis?

Accepted Answer

$$\int_{0}^{4} 2\pi x (64 - (16x - x^2$)) \, dx$$

Question 16

Which integral represents the area of the region bounded by $$y = \sqrt{x}$$ and $$y = x^3$ in the first quadrant?

Accepted Answer

$$\int_{0}^{1} (\sqrt{x} - x^3$) \, dx$$

Question 17

Which integral represents the work required to empty a cone-shaped tank (height $$9 ft$$, top radius $$6 ft) by $$pumping water over the top edge?

Accepted Answer

$$\int_{0}^{9} 62.4 \pi \left(\frac{2}{3} y \$$right)^2$$ (9 - y) \, dy$$

MAT 2420 Calculus II Exam 1 Study Guide – Step-by-Step Guidance

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