# Here's what students ask on Applications of integrals

Applications of integrals

### Evaluate the following integral. $$\displaystyle\int{2}{x}{\left({1}-{x}^{{-{3}}}\right)}{\left.{d}{x}\right.}$$

Applications of integrals

### Evaluate the following definite integral. $$\displaystyle{\int_{{{0}}}^{{\pi}}}{x}{\sin{{x}}}{\left.{d}{x}\right.}$$

Applications of integrals

### Use areas to evaluate the integral. $$\displaystyle{\int_{{{a}}}^{{{11}{b}}}}{2}{s}\ {d}{s},{0}{<}{a}{<}{b}$$

Applications of integrals

### Evaluate the integral. $$\displaystyle\int\frac{{{3}-{x}}}{{\sqrt{{x}}}}{\left.{d}{x}\right.}$$

Applications of integrals

### Consider the integral as attached, To determine the convergence or divergence of the integral, how many improper integrals must be analyzed? What must be true of each of these integrals for the given integral to converge? $$\displaystyle{\int_{{0}}^{{3}}}\frac{{10}}{{{x}^{{2}}-{2}{x}}}{\left.{d}{x}\right.}$$.

Applications of integrals

### Evaluate the following definite integral. $$\displaystyle{\int_{{{0}}}^{{{\frac{{\pi}}{{{2}}}}}}}{x}{\cos{{2}}}{x}{\left.{d}{x}\right.}$$

Applications of integrals

### Evaluate the integral. $$\displaystyle\int{8}{\sin{{\left({4}{t}\right)}}}{\sin{{\left(\frac{{t}}{{2}}\right)}}}{\left.{d}{t}\right.}$$

Applications of integrals

### Applications of double integrals: A lamina occupies the part of the disk $$\displaystyle{x}^{{2}}+{y}^{{2}}\le{64}$$ in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.

Applications of integrals

### given $$\displaystyle{y}=\frac{{1}}{{x}}$$ is a solution $$\displaystyle{2}{x}^{{2}}{d}{2}\frac{{y}}{{\left.{d}{x}\right.}}+{x}\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}-{3}{y}={0},{x}{>}{0}$$ a) Find a linearly independent solution by reduction the order approach b) Show that 2 solutions are linearly independent c) Write a general solution

Applications of integrals

### Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. $$\displaystyle{y}={5}\frac{{\left({16}−{x}^{{2}}\right)}^{{1}}}{{2}}$$ , y = 0, x = 2, x = 3, about the x-axis

Applications of integrals

### Evaluate the integral using the indicated substitution. $$\displaystyle\int{\cot{{x}}}{\cos{{e}}}{c}^{{{2}}}{x}{\left.{d}{x}\right.},\ {u}={\cot{{x}}}$$

Applications of integrals

### Solve the integral. $$\displaystyle\int{x}{\ln{{x}}}{\left.{d}{x}\right.}$$

Applications of integrals

### Evaluate the integral using the indicated substitution. $$\displaystyle\int{{\cos}^{{{2}}}{x}}{\sin{{x}}}{\left.{d}{x}\right.},\ {u}={\cos{{x}}}$$

Applications of integrals

### Suppose $$\displaystyle{\int_{{5}}^{{6}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={6}{\quad\text{and}\quad}{\int_{{5}}^{{6}}}{g{{\left({x}\right)}}}{\left.{d}{x}\right.}={2}$$. Evaluate $$\displaystyle{\int_{{5}}^{{6}}}{\left({4}{f{{\left({x}\right)}}}-{2}{g{{\left({x}\right)}}}\right)}{\left.{d}{x}\right.}$$. Remember to include a "+ C"if appropriate.

Applications of integrals

### Right, or wrong? Say which for each formula and give a brief reason for each answer. $$\displaystyle\int{\left({2}{x}+{1}\right)}{2}{\left.{d}{x}\right.}={\left({2}{x}+{1}\right)}\frac{{3}}{{3}}+{C}$$

Applications of integrals

### Trigonometric integral Evaluate the following integrals. $$\displaystyle\int{{\sin}^{{2}}{0}}{{\cos}^{{5}}{0}}{d}{0}$$

Applications of integrals

### Evaluate the following definite integrals: $$\displaystyle{\int_{{0}}^{{1}}}{\left({x}{e}^{{-{x}^{{2}}+{2}}}\right)}{\left.{d}{x}\right.}$$

Applications of integrals

### Evaluate the following definite integrals $$\displaystyle{\int_{{{0}}}^{{{1}}}}{x}{e}^{{{\left(-{x}^{{{2}}}+{2}\right)}}}{\left.{d}{x}\right.}$$

Applications of integrals