Chapter 11

Find the indefinite integral. $$\int 3 x^{-2 / 3} d x$$

Sinclair wishes to supplement his retirement income by $$\$ 300 /$$ month for the next 10 yr. He plans to obtain a reverse annuity mortgage (RAM) on his home to meet this need. Estimate the amount of the mortgage he will require if the prevailing interest rate is \(8 \% /\) year compounded continuously.

Sketch the graphs of the functions \(f\) and \(g\) and find the area of the region enclosed by these graphs and the vertical lines \(x=a\) and \(x=b\). $$f(x)=-x^{2}+2 x+3, g(x)=-x+3 ; a=0, b=2$$

Evaluate the definite integral. $$\int_{-1}^{1} x e^{x^{2}+1} d x$$

Evaluate the definite integral. $$\int_{1}^{3}(2 x+3) d x$$

Find the indefinite integral. $$\int e^{-2 x} d x$$

Find the indefinite integral. $$\int \frac{2}{x^{2}} d x$$

Sketch the graphs of the functions \(f\) and \(g\) and find the area of the region enclosed by these graphs and the vertical lines \(x=a\) and \(x=b\). $$f(x)=9-x^{2}, g(x)=2 x+3 ; a=-1, b=1$$

Evaluate the definite integral. $$\int_{0}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} d x$$

Evaluate the definite integral. $$\int_{-1}^{0}(4-x) d x$$

Find the indefinite integral. $$\int e^{-0.02 x} d x$$

Find the indefinite integral. $$\int \frac{1}{3 x^{5}} d x$$

A certain country's income distribution is described by the function $$ f(x)=\frac{15}{16} x^{2}+\frac{1}{16} x $$ a. Sketch the Lorentz curve for this function. b. Compute \(f(0.4)\) and \(f(0.9)\) and interpret your results.

Sketch the graphs of the functions \(f\) and \(g\) and find the area of the region enclosed by these graphs and the vertical lines \(x=a\) and \(x=b\). $$f(x)=x^{2}+1, g(x)=\frac{1}{3} x^{3} ; a=-1, b=2$$

Evaluate the definite integral. $$\int_{3}^{6} \frac{2}{x-2} d x$$

Evaluate the definite integral. $$\int_{-1}^{3} 2 x^{2} d x$$

Find the indefinite integral. $$\int e^{2-x} d x$$

Find the indefinite integral. $$\int \pi \sqrt{t} d t$$

In a study conducted by a certain country's Economic Development Board, it was found that the Lorentz curve for the distribution of income of college teachers was described by the function $$ f(x)=\frac{13}{14} x^{2}+\frac{1}{14} x $$ and that of lawyers by the function $$ g(x)=\frac{9}{11} x^{4}+\frac{2}{11} x $$ a. Compute the coefficient of inequality for each Lorentz curve. b. Which profession has a more equitable income distribution?

Sketch the graphs of the functions \(f\) and \(g\) and find the area of the region enclosed by these graphs and the vertical lines \(x=a\) and \(x=b\). $$f(x)=\sqrt{x}, g(x)=-\frac{1}{2} x-1 ; a=1, b=4$$

Evaluate the definite integral. $$\int_{0}^{1} \frac{x}{1+2 x^{2}} d x$$

Evaluate the definite integral. $$\int_{0}^{2} 8 x^{3} d x$$

Find the indefinite integral. $$\int e^{2 t+3} d t$$

Find the indefinite integral. $$\int \frac{3}{\sqrt{t}} d t$$

A certain country's income distribution is described by the function $$ f(x)=\frac{14}{15} x^{2}+\frac{1}{15} x $$ a. Sketch the Lorentz curve for this function. b. Compute \(f(0.3)\) and \(f(0.7)\).

Sketch the graphs of the functions \(f\) and \(g\) and find the area of the region enclosed by these graphs and the vertical lines \(x=a\) and \(x=b\). $$f(x)=\frac{1}{x}, g(x)=2 x-1 ; a=1, b=4$$

Evaluate the definite integral. $$\int_{1}^{2} \frac{x^{2}+2 x}{x^{3}+3 x^{2}-1} d x$$

Evaluate the definite integral. $$\int_{-2}^{2}\left(x^{2}-1\right) d x$$

Find the indefinite integral. $$\int x e^{-x^{2}} d x$$

Find the indefinite integral. $$\int(3-2 x) d x$$

In a study conducted by a certain country's Economic Development Board, it was found that the Lorentz curve for the distribution of income of stockbrokers was described by the function $$ f(x)=\frac{11}{12} x^{2}+\frac{1}{12} x $$ and that of high school teachers by the function $$ g(x)=\frac{5}{6} x^{2}+\frac{1}{6} x $$ a. Compute the coefficient of inequality for each Lorentz curve. b. Which profession has a more equitable income distribution?

Sketch the graphs of the functions \(f\) and \(g\) and find the area of the region enclosed by these graphs and the vertical lines \(x=a\) and \(x=b\). $$f(x)=x^{2}, g(x)=\frac{1}{x^{2}}, a=1, b=3$$

Evaluate the definite integral. $$\int_{0}^{1} \frac{e^{x}}{1+e^{x}} d x$$

Evaluate the definite integral. $$\int_{1}^{4} \sqrt{u} d u$$

Find the indefinite integral. $$\int x^{2} e^{x^{3}-1} d x$$

Find the indefinite integral. $$\int\left(1+u+u^{2}\right) d u$$

Sketch the graphs of the functions \(f\) and \(g\) and find the area of the region enclosed by these graphs and the vertical lines \(x=a\) and \(x=b\). $$f(x)=e^{x}, g(x)=\frac{1}{x} ; a=1, b=2$$

Evaluate the definite integral. $$\int_{1}^{2}\left(4 e^{2 u}-\frac{1}{u}\right) d u$$

Evaluate the definite integral. $$\int_{1}^{8} 4 x^{1 / 3} d x$$

Find the indefinite integral. $$\int\left(e^{x}-e^{-x}\right) d x$$

Find the indefinite integral. $$\int\left(x^{2}+x+x^{-3}\right) d x$$

Sketch the graphs of the functions \(f\) and \(g\) and find the area of the region enclosed by these graphs and the vertical lines \(x=a\) and \(x=b\). $$f(x)=x, g(x)=e^{2 x} ; a=1, b=3$$

Evaluate the definite integral. $$\int_{1}^{2}\left(1+\frac{1}{x}+e^{x}\right) d x$$

Evaluate the definite integral. $$\int_{1}^{4} 2 x^{-3 / 2} d x$$

Find the indefinite integral. $$\int\left(e^{2 x}+e^{-3 x}\right) d x$$

Find the indefinite integral. $$\int\left(0.3 t^{2}+0.02 t+2\right) d t$$

Sketch the graph and find the area of the region bounded by the graph of the function \(f\) and the lines \(y=0, x=a\), and \(x=b\) $$f(x)=x ; a=-1, b=2$$

Evaluate the definite integral. $$\int_{1}^{2}\left(2 e^{-4 x}-\frac{1}{x^{2}}\right) d x$$

Evaluate the definite integral. $$\int_{0}^{1}\left(x^{3}-2 x^{2}+1\right) d x$$

Find the indefinite integral. $$\int \frac{e^{x}}{1+e^{x}} d x$$