# Some double integrals

Homework Helper

## Homework Statement

1) By evaluating a double integral find the volume of the tetrahedron bounded by planes x+2y+z=2, x= 2y ,z=0 and y=0 (answer = 1/3)

2)

$$\int_{0} ^{1} \int_{x} ^{1} sin(y^2) dydx$$

i) explain why this can be evaluated by integrating wrt y first.
ii) sketch the region of integration R of the double integration
iii)Evaluate the integral by expressing it as a Type II region first.

## The Attempt at a Solution

My query is with 1 and 2ii)

for 1

my region I integrated over was x/2≤y≤(2-x)/2 and 0≤x≤1

found when z=0 to get y=(2-x)/2 and this intersects y=x/2 at (1,1/2)

yet when I compute

$$\int_{0} ^{1} \int_{\frac{x}{2}} ^{\frac{2-x}{2}} (2-2y-x) dydx$$

I don't get 1/3. If it's a problem with my algebra I'll fix it. But are my boundaries correct?

2) I believe it can't be done with dy first since the integral of sin(y2) can't be expressed in terms of elementary functions. Is the reason or is there some other factor that prohibits this?

LCKurtz
Homework Helper
Gold Member

And your answer to 2 looks correct too. And that problem will go away when you reverse the integration order.

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Homework Helper
ok for some reason I can't seem to get 1/3. This is what I have

$$\int_{0} ^{1} \int_{\frac{x}{2}} ^{\frac{2-x}{2}} (2-2y-x)dydx$$

$$= \int_{0} ^{1} \left[ 2y-y^2 -xy \right] _{\frac{x}{2}} ^{\frac{2-x}{2}} dx$$

$$= \int_{0} ^{1} \left( (2-x)- \frac{1}{4}(2-x)^2 -\frac{x}{2}(2-x) -x + \frac{x^2}{4} + \frac{x^2}{2} \right)$$

$$= \int_{0} ^{1} (2-x-1+x-\frac{x^2}{4}-x+\frac{x^2}{2}-x+\frac{x^2}{4}+\frac{x^2}{2}) dx$$

$$= \int_{0} ^{1} (1-2x+\frac{x^2}{2}) dx$$

$$= \left[ x-x^2 +\frac{x^3}{6} \right]_{0} ^{1} = 1/6$$

where am I going wrong? Because I clearly can't see it

EDIT: for the second one as well, I can't get the answer, I must be doing something wrong

$$\int_{0} ^{1} \int_{x} ^{1} siny^2 dxdy$$

$$=\int_{0} ^{1} \left[ xsiny^2 \right] _{x} ^{1} dy$$

$$=\int_{0} ^{1} (xsin1-xsinx^2) dx$$

$$= \left[ \frac{x^2}{2}sin1 + \frac{1}{2}cos(x^2) \right] _{0} ^{1}$$

$$= \frac{1}{2}sin1 + \frac{1}{2}cos1 - \frac{1}{2}$$

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LCKurtz
Homework Helper
Gold Member
It's late here so I will be brief. Your limits are correct on the first one and Maple checks it as 1/3. Recheck your algebra.

On the second one your reversed limits are wrong. You need to draw a picture of the region from the first integral and use it to get the dx dy limits.

Sack time here.

Homework Helper
Thanks for your help, I just need a bit of understanding with the 2nd one still. I found the new limits as y≤x≤0 and 0≤y≤1, but to get the answer the y range must be 1≤y≤0 and this makes no sense to me.

LCKurtz
Homework Helper
Gold Member
Thanks for your help, I just need a bit of understanding with the 2nd one still. I found the new limits as y≤x≤0 and 0≤y≤1, but to get the answer the y range must be 1≤y≤0 and this makes no sense to me.

Agreed, 1≤y≤0 makes no sense. Again, to work the problem you must draw the region described in the original integral. You should see a triangular region in the first quadrant above the line y = x and below the line y = 1. Then simply use that picture to get the limits integrating in the x direction first. You need to look at the picture to get the new limits.

Homework Helper
Agreed, 1≤y≤0 makes no sense. Again, to work the problem you must draw the region described in the original integral. You should see a triangular region in the first quadrant above the line y = x and below the line y = 1. Then simply use that picture to get the limits integrating in the x direction first. You need to look at the picture to get the new limits.

I got out this one, but in my integration working above can you see any error I am making because I keep getting answers which are not 1/3 . I just tried it again and got an answer of -1/2

LCKurtz