Integral Problem Solving Made Easy: Learn with Examples

Evaluate the line integral, where C is the given curve
C $(\frac{x}{y})ds,C:x=t^3,y=t^4,1\le t\le 2$

The integral represents the volume of a solid. Describe the solid.
$\pi {\int }_0^1(y^4-y^8)dy$
a) The integral describes the volume of the solid obtained by rotating the region $R=\{\{x,y\}|0\le y\le 1,y^4\le x\le y^2\}$ of the xy-plane about the x-axis.
b) The integral describes the volume of the solid obtained by rotating the region $R=\{\{x,y\}|0\le y\le 1,y^2\le x\le y^4\}$ of the xy-plane about the x-axis.
c) The integral describes the volume of the solid obtained by rotating the region $R=\{\{x,y\}|0\le y\le 1,y^4\le x\le y^2\}$ of the xy-plane about the y-axis.
d) The integral describes the volume of the solid obtained by rotating the region $R=\{\{x,y\}|0\le y\le 1,y^2\le x\le y^4\}$ of the xy-plane about the y-axis.
e) The integral describes the volume of the solid obtained by rotating the region $R=\{\{x,y\}|0\le y\le 1,y^4\le x\le y^8\}$ of the xy-plane about the y-axis.

If f and g are the function whose graph are shown let $u(x)=f(x)g(x)andv(x)=$

Evaluate the line integral, where C is the given curve
${\int }_C(\frac{x}{y})ds,C:x=t^3,y=t^4,1\le t\le 4$

Evaluate the line integral, where C is the given curve
C xy ds
C: $x=t^2,y=2t,0\le t\le 5$

The velocity function (in meters per second) is given for a particle moving along a line.
$v(t)=3t-8,0\le t\le 3$
(a) Find the displacement.
(b) Find the distance traveled by the particle during the given time interval.

Evaluate the line integral, where C is the given curve.
${\int }_{C}xyds$
$C:x=t^2,$
$y=2t,$
$0\le t\le 4$

$x=\sin \left(\frac{\theta }{2}\right),y=\cos \left(\frac{\theta }{2}\right),-\pi \le \theta \le \pi$ (a) Eliminate the parameter to find a Cartesian equation of the curve. and how does thecurve look

Evaluate the double integral. double integral $(x^2+2y)dA$, D is bounded by $y=x,y=x^3,x\ge 0$

Evaluate the integral. (Use C for the constant of integration.)
$\int \ln (\sqrt{x})dx$

Suppose that $f(x)=0,125x\text{ for }0$. Determine the mean and variance of X.

Find a Cartesian equation for the curve and identify it.
$r^2\cos 2\theta =1$

A variable force of ${\displaystyle 9x-29x-2}$ pounds moves an object along a straight line when it is xx feet from the origin. Calculate the work done in moving the object from ${\displaystyle x=1ft,\to ,x=19ft.}$ (Round your answer to two decimal places.)

Evaluate the integral ${\displaystyle \int e^{3x}\cos 2xdx}$

Show by substitution that $u(x,t)=\cos (\alpha \pi x)e^{-}{\alpha }^2{\pi }^{2}t$ is a solution of the heat equation $ut=uxx$ on any interval [0, L].

Compute ${\displaystyle {\int }_0^2\frac{1}{x-1}dx}$

A particle with velocity at any time t is given by ${\displaystyle v\left(t\right)=2e^{2t}}$ moves in a straight line. How far does the particle travel during the time interval when its velocity increases from 2 to 4?

Give the correct answer and solve the given equation $[x-y\arctan (\frac{y}{x})]dx+x\arctan (\frac{y}{x})dy=0$

Give the correct answer and solve the given equation $dx+(\frac{x}{y}-\sin y)dy=0$

$(x^2+2xy-4y^2)dx-(x^2-8xy-4y^2)dy=0$