💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!
Section 2
Areas by Integration
Make the given changes in the indicated examplesof this section and then find the resulting areas.In Example $1,$ change $x=2$ to $x=3$
Make the given changes in the indicated examplesof this section and then find the resulting areas.In Example $3,$ change $y=x+2$ to $y=2 x$
Find the areas bounded by the indicated curves.$$y=4 x, y=0, x=1$$
Find the areas bounded by the indicated curves.$$y=3 x+3, y=0, x=2$$
Find the areas bounded by the indicated curves.$$y=8-2 x^{2}, y=0$$
Find the areas bounded by the indicated curves.$$y=\frac{1}{2} x^{2}+2 ; x=0, y=4(x>0)$$
Find the areas bounded by the indicated curves.$$y=x^{2}-4, y=0, x=-4$$
Find the areas bounded by the indicated curves.$$y=4 x^{2}-6 x, y=0$$
Find the areas bounded by the indicated curves.$$y=3 / x^{2}, y=0, x=2, x=3$$
Find the areas bounded by the indicated curves.$$y=16-x^{2}, y=0, x=-2, x=3$$
Find the areas bounded by the indicated curves.$$y=4 \sqrt{x}, x=0, y=1, y=3$$
Find the areas bounded by the indicated curves.$$y=3 \sqrt{x+1}, x=0, y=6$$
Find the areas bounded by the indicated curves.$$y=2 / \sqrt{x}, x=0, y=1, y=4$$
Find the areas bounded by the indicated curves.$$x=y^{2}-4 y, x=0$$
Find the areas bounded by the indicated curves.$$y=6-3 x, x=0, y=0, y=3$$
Find the areas bounded by the indicated curves.$$y=x, y=3-x, x=0$$
Find the areas bounded by the indicated curves.$$y=x-4 \sqrt{x}, y=0$$
Find the areas bounded by the indicated curves.$$y=2 x^{3}-x^{4}, y=0$$
Find the areas bounded by the indicated curves.$$y=x^{2}, y=2-x, x=0 \quad(x \geq 0)$$
Find the areas bounded by the indicated curves.$$y=x^{2}, y=2-x, y=1$$
Find the areas bounded by the indicated curves.$$y=x^{4}-8 x^{2}+16, y=16-x^{4}$$
Find the areas bounded by the indicated curves.$$y=\sqrt{x-1}, y=3-x, y=0$$
Find the areas bounded by the indicated curves.$$y=x^{2}+5 x, y=3-x^{2}$$
Find the areas bounded by the indicated curves.$$y=x^{3}, y=x^{2}+4, x=-1$$
Find the areas bounded by the indicated curves.$$y=\frac{1}{2} x^{5}, x=-1, x=2, y=0$$
Find the areas bounded by the indicated curves.$$y=x^{2}+2 x-8, y=x+4$$
Find the areas bounded by the indicated curves.$$y=4-x^{2}, y=4 x-x^{2}, x=0, x=2$$
Find the areas bounded by the indicated curves.$$y=x^{2} \sqrt{1-x^{3}}, y=0$$
Describe a region for which the area is found by evaluating the integral $\int_{1}^{2}\left(2 x^{2}-x^{3}\right) d x$
Solve the given problems.Although the integral $\int_{-2}^{2} \sqrt{4-x^{2}} d x$ cannot be integrated by methods we have developed to this point, by recognizing the region represented, it can be evaluated. Evaluate this integral.
Solve the given problems.Use integration to find the area of the triangle with vertices (0,0)(4, 4), and (10, 0).
Solve the given problems.Show that the area bounded by the parabola $y=x^{2}$ and the line $y=b(b>0)$ is two-thirds of the area of the rectangle that circumscribes it.
Solve the given problems.Show that the curve $y=x^{n}(n>0)$ divides the unit square bounded by $x=0, y=0, x=1,$ and $y=1$ into regions with areas in the ratio of $n / 1$
Solve the given problems.Why can the integral $\int_{a}^{2}\left(2+x-x^{2}\right) d x$ be used to find the area bounded by $x=a, y=0,$ and $y=2+x-x^{2}$ if $a=-1,$ but not if $a=-2 ?$
Solve the given problems.Find the area of the parallelogram with vertices at (0,0),(2,0)(2, 1), and (4, 1) by integration. Show any integrals you set up.
Solve the given problems.Set up the integrals (do not evaluate) for the upper of the two areas bounded by $y=4-x^{2}, y=3 x,$ and $y=4-2 x,$ using vertical elements of area.
Solve the given problems.Find the value of $c$ such that the region bounded by $y=x^{2}$ and $y=4$ is divided by $y=c$ into two regions of equal area.
Solve the given problems.Find the positive value of $c$ such that the region bounded by $y=x^{2}-c^{2}$ and $y=c^{2}-x^{2}$ has an area of 576
Find the areas bounded by the indicated curves, using (a) vertical elements and (b) horizontal elements.$$y=8 x, x=0, y=4$$
Find the areas bounded by the indicated curves, using (a) vertical elements and (b) horizontal elements.$$y=x^{3}, x=0, y=3$$
Find the areas bounded by the indicated curves, using (a) vertical elements and (b) horizontal elements.$$y=4 x, y=x^{3}$$Find the areas bounded by the indicated curves, using (a) vertical elements and (b) horizontal elements.$$y=x^{4}, y=8 x$$
Find the areas bounded by the indicated curves, using (a) vertical elements and (b) horizontal elements.$$y=4 x, y=x^{3}$$
Certain physical quantities are often represented as an area under a curve. By definition, power is the time rate of change of performing work. Thus, $p=d w / d t$ or $d w=p d t .$ If $p=12 t-4 t^{2},$ find the work (in J) performed in 3 s by finding the area under the curve of $p$ vs. $t .$ See Fig. $26.16 .$ Round the answer to three significant digits.
The total electric charge $Q$ (in $\mathrm{C}$ ) to pass a point in the circuit from time $t_{1}$ to $t_{2}$ is $Q=\int_{t_{1}}^{t_{2}} i d t,$ where $i$ is the current (in A). Find $Q$if $t_{1}=1 \mathrm{s}, t_{2}=4 \mathrm{s},$ and $i=0.0032 t \sqrt{t^{2}+1}$
Because the displacement $s,$ velocity $v,$ and time $t$ of a moving object are related by $s=\int v d t$, it is possible to represent the change in displacement as an area. A rocket is launched such that its vertical velocity $v$ (in $\mathrm{km} / \mathrm{s}$ ) as a function of time $t$ (in s) is $v=1-0.01 \sqrt{2 t+1} .$ Find the change in vertical displacement from $t=10 \mathrm{s}$ to $t=100 \mathrm{s}$
The total cost $C$ (in dollars) of production can be interpreted as an area. If the cost per unit $C^{\prime}$ (in dollars per unit) of producing $x$ units is given by $100 /(0.01 x+1)^{2}$, find the total cost of producing 100 units by finding the area under the curve of $C^{\prime}$ vs $x$.
A cam is designed such that one face of it is described as being the area between the curves $y=x^{3}-2 x^{2}-x+2$ and $y=x^{2}-1$ (units in $\mathrm{cm}$ ). Show that this description does not uniquely describe the face of the cam. Find the area of the face of the cam, if a complete description requires that $x \leq 1$
Using CAD (computer-assisted design), an architect programs a computer to sketch the shape of a swimming pool designed between the curves $$y=\frac{800 x}{\left(x^{2}+10\right)^{2}} \quad y=0.5 x^{2}-4 x \quad x=8$$ (dimensions in $m$ ). Find the area of the surface of the pool.
A coffee-table top is designed to be the region between $y=0.25 x^{4}$ and $y=12-0.25 x^{4}$ (dimensions in $\mathrm{dm}$ ). What is the area of the table top?
A window is designed to be the area between a parabolic section and a straight base, as shown in Fig. 26.17. What is the area of the window?