Exam 13: Integral Calculus

Work the exercise. -In a certain type of specialty store, the rate of sales (in billions of dollars per year) is approximately $f(x)=1.72 \ln (x)+16(x \geq 1)$ , where $x=1$ corresponds to the year 2001. Find the function $F(x)$ that gives total sales between the year 2001 and year $x$ .

Use the definite integral to find the area between the $x$ -axis and the graph of $f(x)$ over the indicated interval. - $f(x)=x^{2}+1 ;[0,1]$

Use the definite integral to find the area between the $x$ -axis and the graph of $f(x)$ over the indicated interval. - $f(x)=2 x-x^{2} ;[0,2]$

Work the exercise. -The rate of sales at a footwear outlet (in thousands of dollars per day) can be approximated by the function $f(x)=8 x e^{-x / 7}$ , where $x$ is in days and $x=0$ corresponds to the start of October. (a) Find the function $F(x)$ that gives total sales since the start of October through day $x$ . (Hint: $\mathrm{F}(0)=0$ .) (b) Find the total sales since the start of October through $x=4$ . Round your answer to the nearest dollar, if necessary.

Solve the problem. -A force acts on a certain object in such a way that when the object has moved a distance of $r$ (in $\mathrm{m}$ ), the force $\mathrm{f}$ (in newtons) is given by $\mathrm{f}=3 \mathrm{r}^{2}+2 \mathrm{r}$ . Find the work (in joules) done through the first four meters.

Solve the problem. -A certain company has found that its expenditure rate per day (in hundreds of dollars) on a certain type of job is given by $\mathrm{E}^{\prime}(\mathrm{x})=6 \mathrm{x}+11$ , where $\mathrm{x}$ is the number of days since the start of the job. Find the expenditure if the job takes 6 days.

Find the integral. - $\int x^{4}\left(x^{3}+12 x-7\right) d x$

Find the integral. - $\int \frac{x^{5}}{x^{6}+7} d x$

Solve the problem. -A particle moves so that its velocity (in $\mathrm{m} / \mathrm{s}$ ) is given by $\mathrm{v}=2 \mathrm{te}^{-\mathrm{t}}$ , where $\mathrm{t}$ is the time (in seconds). Find the distance traveled between $\mathrm{t}=0$ and $\mathrm{t}=5$ .

Approximate the area under the curve and above the $x$ -axis using $n$ rectangles. Let the height of each rectangle be given by the value of the function at the right side of the rectangle. - $f(x)=3 x^{2}-2$ from $x=1$ to $x=5 ; n=4$

Find the integral. - $\int \frac{x}{(1-x)^{4}} d x$

Provide the proper response. -When we define the definite integral, what is a necessary condition for the function $f(x)$ ?

Approximate the area under the curve and above the $x$ -axis using $n$ rectangles. Let the height of each rectangle be given by the value of the function at the right side of the rectangle. - $f(x)=2 x+3$ from $x=0$ to $x=2 ; n=4$

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

Find the area bounded by the given curves. - $y=x^{3}, y=x^{2}$

Solve the problem. -The current (in amperes) in an inductor of inductance $\mathrm{L}$ (in henries) is given by $\mathrm{i}=\frac{1}{\mathrm{~L}} \int \mathrm{Vdt}$ , where $\mathrm{V}$ is the voltage (in volts) and $\mathrm{t}$ is the time (in seconds). Find a formula for $\mathrm{i}$ , if $\mathrm{V}=7 \mathrm{t}\left(\mathrm{t}^{2}-4\right)$ .

Solve the problem. -Find the cost function if the marginal cost function is $C^{\prime}(x)=18 x-5$ and the fixed cost is $\$ 3$ .

Evaluate using integration by parts. - $\int(2 x-1) \ln (21 x) d x$

Find the integral. - $\int t e^{\left(-7 t^{2}\right)} d t$

Evaluate. - $\int\left(2 x^{5}-7 x^{3}+6\right) d x$