Calculus 2: test 1

Grading

This is a set of exercises for Calculus 2: course.

1. Sketch the region bounded by the graphs of the equations $x=y^{2}+1,x=y+3$ and determine the area of the region.\newpage
2. The region bounded by the graphs of $y=2\sqrt{x},y=0,$ and $x=3$ is revolved about the $x$-axis. Find the surface area of the solid generated.
3. (a) Set up the Riemann sum for the volume of the solid obtained by revolving the region under the graph of a continuous function $y=f(x)\geq 0,a\leq x\leq b,$ about the $x$-axis, provide an illustration and the integral formula.(b) Use the formula to set up the integral for the volume of the circular cone of radius $1$ and height $1$ (do not evaluate).
4. Evaluate the indefinite integral $$\int \dfrac{\cos t}{\sin^{2}t+2\sin t+1}dt.$$
5. Evaluate the indefinite integral $$\int \sqrt{1-y^{2}}dy.$$
6. Evaluate the indefinite integral $$\int \dfrac{x}{(x+1)(x^{2}+1)}dx.$$
7. Evaluate the indefinite integral $$\int xe^{2x}dx.$$

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1. Set up the Riemann sum for the volume of the sphere of radius $R$, provide an illustration and the integral formula.
2. The region bounded by the graphs of $y=2\sqrt{x},y=0,$ and $x=3$ is revolved about the $x$-axis. Find the surface area of the solid generated.
3. A cable that weighs 2 lb/ft is used to lift 800 lb of coal up a mineshaft 500 ft deep. Find the work done.
4. Evaluate the indefinite integral $$\int\sin^{3}t\cos^{3}tdt.$$
5. Evaluate the indefinite integral $$\int\dfrac{dx}{x^{2}+4}.$$
6. Find the partial fraction decomposition for the indefinite integral \[\int\dfrac{x}{(x+1)(x^{2}+1)}dx.\] Evaluate it.
7. Evaluate the indefinite integral $$\int\dfrac{\ln x}{x^{2}}dx.$$

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1. Set up the Riemann sum for the area of the circle of radius $R$ as the area between two curves, provide an illustration and the integral formula. Evaluate for extra 5 points.
2. The region bounded by the graphs of $y=x^{2}+1,y=0,x=0,$ and $x=1$ is revolved about the $x$-axis. Find the volume area of the solid generated.
3. An aquarium 2 m long, 1 m wide, and 1 m deep is full of water. Find the work needed to pump half of the water out of the aquarium (the density of water is 1000 kg/m$^{3})$.
4. Evaluate the indefinite integral $$\int x^{2}\sin xdx.$$
5. Evaluate the indefinite integral $$\int \dfrac{dx}{4-x^{2}}.$$
6. Find the partial fraction decomposition for the indefinite integral $$\int \dfrac{x^{5}-1}{x^{3}(x+1)(x^{2}+1)^{2}}dx.$$ Do not evaluate the coefficients.
7. Evaluate the indefinite integral $$\int \dfrac{\sin x}{1+\cos^{2}x}dx.$$