Applications of Integrals
(Thomas-Finney Section 5.1, page 372)
Problem 31
Find the area of the region enclosed by the curves and .
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First, we'll plot the region of interest. Since the equations are easily solved for y, we'll do that first:
> | eqn1:=4*x^2+y=4; eqn2:=x^4-y=1; |
> | y1:=solve(eqn1,y); y2:=solve(eqn2,y); |
And we'll solve for where the curves cross so we know where to plot:
> | solve({eqn1,eqn2},{x,y}); |
The third thing is complex, so looks like x should go from -1 to 1. We'll go a little farther, just in case:
> | plot({y1,y2},x=-2..2,color=blue,thickness=2); |
Now we're ready to calculate the area. y1 is on top since it's the parabola that opens down:
> | area:=int(y1-y2,x=-1..1); |
Section 5.3, page 385 Problem 9
Find the volume of the solid generated by revolving the region bounded by
, for x in , ,
about the x axis.
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First we'll graph the region and the solid:
> | with(plots,tubeplot): |
> | plot(sqrt(cos(x)),x=0..Pi/2,color=blue,thickness=2); |
> | tubeplot([x,0,0],x=0..Pi/2,radius=sqrt(cos(x))); |
To calculate the volume, we just integrate from 0 to :
> | volume:=int(Pi*(sqrt(cos(x))^2),x=0..Pi/2); |