Section Practice
¶These are practice problems for the tests. Solutions are in the next section. While we will not present these, I am happy to answer questions about them in class.
Riemann Sums
Sketch \(f(x)=x^{2}+2\) on \([0,3].\) Subdivide the interval into 6 subintervals of equal length and compute the upper and lower Riemann sums of \(f\) over this partition.
Use the definition of the integral (as the limit of Riemann sums) to compute the area under \(f(x)=x^{2}+2\) on \([0,3].\)
Approximate the area under \(f(x)=-\sqrt{5-x}\) on \([1,2]\) with \(n=4\) using upper and lower sums.
Definite and Indefinite Integrals
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Compute the following indefinite integrals.
\(\dsp{\int 2x+ \frac{3}{4} x^2 \; dx}\)
\(\dsp\int 1+e^s \; ds\)
\(\dsp \int \sin(x) \; dx\)
\(\dsp \int \cos(y) \; dy\)
\(\dsp \int \pi \cos(x) + \pi \; dx\)
\(\dsp \int \sin(2t) \; dt\)
\(\dsp{\int 5\sqrt{x}-\frac{1}{x} \; dx}\)
\(\dsp \int (x^2 + 1) \; dx\)
\(\dsp \int (x^2 + 1)^2 \; dx\)
\(\dsp \int x(x^2 + 1)^2 \; dx\)
\(\dsp \int 2x\cos(x^2) \; dx\)
\(\dsp{ \int \frac{5}{t} +\frac{1}{5t}+5^{t}\; dt}\)
\(\dsp \int \sin^4(t)\cos(t) \; dx\)
\(\dsp \int (5x^3 - 3x^2)^8(15x^2 - 6x) \; dx\)
\(\dsp \int (y^2-2y)^2(5y^4-2) \; dy\)
\(\dsp \int t^2(t^3 - 3t) \; dt\)
\(\int -\pi x + e \; dx\)
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Compute the following definite integrals.
\(\dsp \int_{0}^{2} 6-2x \; dx\)
\(\dsp \int_{2}^{5} x^{2}-6x+10 \; dx\)
\(\dsp \int_{-1}^{3} 4x^{3}-x+2 \; dx\)
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Evaluate each integral.
\(\dsp \int 2x+1 \ dx\)
\(\dsp \int \frac{1}{3}t^{3}-\sqrt{t} \ dt\)
\(\dsp \int 4(\sec^{2}(x)-\cos(x)) \ dx\)
\(\dsp \int z\sqrt{z}+\frac{4}{z^{2}}+\sqrt[3]{4z} \ dz\)
\(\dsp \int \frac{x+1}{\sqrt{2x}} \ dx\)
\(\dsp \int \sin(t)+\sqrt{\frac{3}{t}} \ dt\)
\(\dsp \int_{}^{} x^{2}\sin(x^{3}) \ dx\)
\(\dsp \int_{}^{} \cos^{5}(x)\sin(x) \ dx\)
\(\dsp \frac{1}{2} \int_{}^{} 2x\sqrt{1-x^{2}} \ dx\)
\(\dsp \int \frac{z+1}{^{3}\sqrt{3z^{2}+6z+5}} \ dz\)
\(\dsp \int_{-1}^{0} (\cos(2t)+t) \ dt\)
\(\dsp \int_{0}^{\frac{\pi}{4}} \sin(\theta )-\cos(\theta ) \ d\theta\)
\(\dsp \int_{-2}^{1} (5-x)^{3} \ dx\)
\(\dsp \int_{\pi /4}^{\pi /2} \frac{\cos(x)}{\sin^{2}(x)} \ dx\)
\(\dsp \int_{-\sqrt{3}}^{2} 4x\sqrt{x^{2}+1} \ dx\)
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Evaluate each of the following integrals, utilizing the suggested substitution.
\(\dsp \int \frac{x}{\sqrt{x+5}} \; dx\text{;}\) \(u(x) = \sqrt{x+5}.\)
\(\dsp \int x \sqrt[3]{x-1} \; dx\text{;}\) \(u(x) = \sqrt[3]{x-1}.\)
\(\dsp \int \frac{x^3}{\sqrt{4-x^2}} \; dx\text{;}\) \(u(x) = \sqrt{4-x^2}.\)
\(\dsp \int \frac{x^3}{\sqrt{x^2+1}} \; dx\text{;}\) \(u(x) = \sqrt{x^2+1}.\)
\(\dsp \int \sqrt{9-x^2} \; dx\text{;}\) \(x = 3\sin(u).\)
\(\dsp \int \frac{1}{4-x^2} \; dx\text{;}\) \(x = 2\sin(u).\)
\(\dsp \int \frac{1}{1+x^2} \; dx\text{;}\) \(x = \tan(u).\)
Average Value, Mean Value, Fundamental Theorem and Arclength
Compute the average value of \(f(x) = x^3\) over \([-3,1].\)
Compute the average value \(f(x) = |x|\) over \([-1,1].\)
Let \(f(x) = x^2 + 2x-4\) be the continuous function on \([0,4]\text{.}\) Find \(c\) in \([0,4]\) as stated in the Mean Value Theorem for integrals.
Let \(F(x) =\dsp \int_{1 \over 2}^{x} \cos(t) \; dt\text{.}\) Compute the integral and then take the derivative of your result.
Compute \(\dsp {{d} \over {\; dx}} \int_{1 \over 2}^{x} t^3 - 4t + e^t \; dt\)
Use the chain rule to compute \(\dsp {{d} \over {\; dx}} \int_{9}^{\sqrt{x}} {e^{t^2}} \; dt\)
Find the arc length of \(y^2=4x^3\) from \((0,0)\) to \((4,16).\)
Area and Volume
Find the area of the region bounded by \(y=x^2\text{,}\) \(x=-2\text{,}\) \(x=-1\text{,}\) and the \(x\)-axis.
Find the area of the region bounded by \(y=2|x|\text{,}\) \(x=-3\text{,}\) \(x=1\text{,}\) and the \(x\)-axis.
Find the area of the region bounded by \(y=x^2-4\) and the \(x\)-axis.
Find the area of the region bounded by \(y=x^2\) and \(y=-x^2+6x\text{.}\)
Find the area of the region bounded by \(y=x^3\) and \(y=4x\text{.}\)
Find the area of the region bounded by \(x=y^2\) and \(x-2y=3\text{.}\)
Is there a region bounded by \(x-3=y^2-4y\) and \(x=-y^2+2x+3\text{?}\)
Find the area of the region bounded by \(x=2y-y^2\) and \(x=-y\text{.}\)
Find the area of the region bounded by \(y=x^3+3x^2-x+1\) and \(y=5x^2+2x+1\text{.}\)
Find the volume of the solid generated by revolving the region bounded by the curve \(y = \sqrt{x}\text{,}\) the \(x\)-axis, and the line \(x=4\) revolving about the \(x\)-axis using (a) the disk method and (b) the shell method.
Find the volume of the solid generated by revolving about the \(y\)-axis the region bounded by the line \(y=2x\) and the curve \(y=x^2\text{.}\)
Find the volume of the solid generated by revolving about the line \(x=6\) the region bounded by the line \(y=2\) and the curve \(y=x^2\text{.}\)
Compute the volume of the solid whose base is the disk \(x^2+y^2 \le 4\) and whose cross sections are equilateral triangles perpendicular to the base.
Verify the formula for the volume of a sphere by revolving the region bounded by the circle \(x^2 + y^2=R^2\) about the \(y\)-axis with the disk method.
Find the volume of the solid generated by revolving the region bounded by the curve \(y=(4x-1)^{1/3}\text{,}\) the \(x\)-axis, and the line \(x=4\text{,}\) revolving about the \(x\)-axis using the (a) cylindrical shell method and (b) the disk method.
Find the volume of the solid generated by revolving the region bounded by the curve \(y^2-y=x\) and the \(y\)-axis, about \(x\)-axis.
Find the volume of the solid generated by revolving the region bounded by the curve \(y=8-x^2\text{,}\) \(y=x^2\text{,}\) about the \(y\)-axis.
Find the volume of the solid generated by revolving the region by the curves \(y=|x|+1\) and \(y=2\text{,}\) about the \(x\)-axis.
Work
Suppose we have a spring which is 6 inches at rest and 10 pounds of force will stretch it 4 inches. Write out and compute an integral that equals the work required to stretch the spring 6 inches from its resting state.
Find the work required for a crane to pull up 500 feet of cable weighing 2 lbs/ft with an 800 pound wrecking ball on it.
Suppose we have a 1 m deep aquarium with a base of 1 m by 2 m which is completely full of water. How much work is required to pump half of the water out?
Center of Mass
Let the length of a rod be 10 meters and the linear density of the rod \(\rho(x)\) be written in the form \(\rho(x)=ax+b\) with \(x=0\) representing the left end of the rod and \(x=10\) representing the right end of the rod. If the density of the rod is \(2 kg/m\) at the left end and 17 kg/m at the right end, find the mass and the center of mass of the rod.
Find the centroid (center of mass with of an object with constant density) of the region bounded by \(y=x^3\) and \(y=x\) in the first quadrant. What would the center of mass be if we considered only the third quadrant? What if we considered both the first and third quadrant?