Double Integral Homework: Find Heat in Metal Plate Region x2+y2≥1

[arl146]

Homework Statement

Consider the heat density function for a metal plate: x + y² + x²

Find the total heat in the plate given that the plate resides in the region x≥0, y≥0, x² + y² ≥ 1

The Attempt at a Solution

i thought it was pretty straightforward. i did a double integral with both limits being 0 to 1 but my professor said the y limit is not to 1. what is the top limit then?

 

[ehild]

Find the total heat in the plate given that the plate resides in the region x≥0, y≥0, x² + y² ≥ 1

Are you sure you copied the last formula correctly? I would mean both x and y going to infinity.

ehild

 

[arl146]

should be less than or equal to 1

 

[arl146]

so is the upper limit of y just sqrt(1-x^2) ?

 

[Quinzio]

Correct

 

[arl146]

ok now i can't do the integral..first step, integrate WRT y. so the function becomes x + (y^3)/3 + x^2
plugging the limits for y in:

[x + [ (1-x^2)^(3/2) / 3 ] + x^2] - x - 0 - x^2

so youre just left with [ (1-x^2)^(3/2) / 3 ]. how do you integrate that WRT x ??

 

[HallsofIvy]

Let $x= sin(\theta)$. That's a pretty standard integral.