Math Archive: Questions from 2024-05-2

For each of the improper integrals below, if the comparison test applies, enter either A or B followed by one letter from C to K that best applies, and if the comparison test does not apply, enter only L. For example, one possible answer is BF, and another one is L. Hint: 0 < e−x ≤ 1 for x ≥ 1. ∫1∞1 x2+5 dx ∫1∞ 10+sin⁡(x) x−0.3 dx ∫1∞ x x6+5 dx ∫1∞ cos2⁡(x) x2+5 dx ∫1∞ e−x x2dx A. The integral is convergent B. The integral is divergent C. by comparison to ∫1∞ 1x2−5 dx. D. by comparison to ∫1∞ 1x2+5 dx. E. by comparison to ∫1∞ cos2⁡(x)x2 dx. F. by comparison to ∫1∞ ex x2 dx. G. by comparison to

Match each improper integral with the improper integral below to which you can compare using the Comparison Test, then determine whether the integrals converge or diverge. A. ∫1∞dxx B. ∫1∞dxx2 C. ∫1∞dxx3, and D. ∫1∞dxxNote: you may need to use some of the integrals A-D more than once, and you might not need to use them all. ? ∫1∞dxx3+4 Does this integral converge or diverge? ?? ∫1∞2+e−xxdx Does this integral converge or diverge? ∫1∞xx3+1dx Does this integral converge or diverge? ∫1∞2+cos⁡xxdx Does this integral converge or diverge? ?

Use any method to determine if the series converges or diverges. Give reasons for your answer. ∑n = 1∞ (−3)n4nSelect the correct choice below and fill in the answer box to complete your choice. A. The series diverges because it is a geometric series with r = . B. The series converges because the limit used in the Ratio Test is . C. The series diverges per the Integral Test because ∫1∞ 14x dx = D. The series converges because it is a p-series with p =

Find the volumes of the solids generated by revolving the region between y = 4x and y = x22 about a) the x-axis and b) the y-axis. The volume of the solid generated by revolving the region between y = 4x and y = x22 about the x-axis is cubic units. (Round to the nearest tenth.) The volume of the solid generated by revolving the region between y = 4x and y = x22 about the y-axis is cubic units. (Round to the nearest tenth.)

We model pumping from spherical containers the way we do from other containers, with the axis of integration along the vertical axis of the sphere. Use the figure shown to the right to find how much work it takes to empty a full hemispherical water reservoir of radius 5m by pumping the water to a height of 3m above the top of the reservoir. Water weighs 9800N/m3. The amount of work required to pump the water to a height of 3m above the top of the reservoir is approximately (Do not round until the final answer. Then round to the nearest hundredth as needed.)

Find the volume of the solid generated by revolving the region bounded by the given curve and lines about the x-axis. y = x, y = 3, x = 0 V = (Type an exact answer, using π as needed. )

Calculate the fluid force on one side of the plate using the coordinate system shown below. Assume the density is 62.4 lb/ft3. The fluid force on one side of the plate is

Find the area of the surface generated when the given curve is revolved about the x-axis. y = 4x+6 on [0, 3] The area of the generated surface is square units. (Type an exact answer, using π as needed.)

Find the volume of the given right tetrahedron. (Hint: Consider slices perpendicular to one of the labeled edges.) V = cubic units

Evaluate the integral using any appropriate algebraic method or trigonometric identity. ∫5dx x 16−25ln2⁡x ∫5dx x16−25ln2⁡x = (Type an exact answer.)

Use the arc length formula to find the length of the line segment y = 3−2x, 0 ≤ x ≤ 3. The length of the line segment is (Type an exact answer, using radicals as needed.)

A schematic drawing of a 118-ft dome used by a weather service to house radar is shown to the right. a. How much outside surface is there to paint (not counting the bottom)? b. Express the answer to the nearest square foot. a. Set up the integral that can be used to find the amount of surface to paint. Select the correct choice below and fill in the answer box to complete your choice. A. S = ∫−29.5 59 dy B. S = ∫−29.5 59 dx The outside surface to paint is (Type an exact answer, using π as needed. ) b. The outside surface to paint is approximately (Round to the nearest integer as needed.)

Use separation of variables to provide implicit solutions to each differential equation. If an initial condition is provided, find the solution satisfying it. (a) dydx = 3x2 3y2−4, y(1) = 0 (c) dydx = (x−3)(y−5), y(4) = 5 (b) dydx = y2 x, y(1) = 2 (d) dydt = t2+et y1+y2

Find and classify (as stable, unstable, or semistable) all equilibrium solutions for the following differential equations. Summarize your work in a phase line. (a) y′ = y(3−y)(25−y2) (b) y′ = ln⁡y−2, y > 0 (c) y′ = 5y+2, y ≥ 0