Class Notes (811,179)
Mathematics (837)
MATA33H3 (103)
Lecture

# Reversing Integration

3 Pages
106 Views

School
University of Toronto Scarborough
Department
Mathematics
Course
MATA33H3
Professor
Eric Moore
Semester
Fall

Description
Reversing the Order of Integration Let z = f(x,y) and let B be a subset of the Xy-plane. The volume of the ﬁgure R lying below the graph of f(x,y) and over the set B is denBtedx,y)dA To compute it requires a double integral. If B is the region between the curves y = g(x) (below) and R RbR h(x) y = h(x) (above) for x between a and b, thBn(x,y)dA = a g(x)f(x,y)dydx. Note that the limits of integration depend only the region B and are independent of the function z = f(x,y) being integrated. Since the preceding discussion is symmetrical with respect to x and y, we cab reverse the roles of x and y if convenient. That is, if B is written instead as the region boundeR by the curves x = p(y) (Rn the left) aR Rx = q(y) (on the right) for y between c and d then f(x,y)dA is also given by f(x,y)dA = d q(yf(x,y)dxdy. B B c p(y) 2 Example. Let f(x,y) R xy + y and let B be the region between the curves y = x and y = x . Compute B f(x,y)dA. Solution 1. The curves y = x and y = x intersect at x = 0 and x = 1. For x between 0 and 1, 2 2 x ≤ x. Therefore B is the region between y = x (below) and y = x (above) for x between 0 and 1. Therefore Z Z Z Z 1 2 1 xy2 y3 y=x f(x,y)dA = (xy + y )dy dx = + dx B 0 x 0 2 3 y=x2 Z 1 Z 1 x3 x3 x5 x6 5x3 x5 x6 = + − − dx = − − dx 0 2 3 2 3 0 6 2 3 4 6 71 = 5x − x − x = 5 − 1 − 1 = 13 24 12 21 0 24 12 21 168 Solution 2. √ Solving for the √urves for x in terms of y gives x = y andy respectivel√. The curves x = y and x = y intersect at y = 0 and y = 1. For y between 0 and 1, y.≤ Therefore Z Z 1Z √ y Z 1 x= y 2 x y 2 f(x,y)dA = (xy + y )dxdy = + xy dy B 0 y 0 2 x=y Z 1 2 3 Z 1 2 = y + y5/2− y − y3 dy = y + y5/2− 3 y3 dy 2 2 2 2 0 0 y3 2y7/2 3 1 1 2 3 13 = − − y4 = − − = 6 7 6 0 6
More Less

Related notes for MATA33H3

OR

Don't have an account?

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Just a few more details

So we can recommend you notes for your school.