Multivariable Questions?

[匿名]

2015-04-06 6:51 am

1. Calculate the double integral of f(x,y) = y^2 over rhombus R (rhombus y = [-4,4], x = [-2,2])

2. Calculate the double integral f(x,y) = (x+y)^-1 bounded by y=x,y=1,y=e, and x=0

3. Calculate the double integral of e^x bounded by lines y=x+1, y=x, x=0, and x=1

回答 (1)

2015-04-06 7:41 am

1) ∫(y = -4 to 4) ∫(x = -2 to 2) y^2 dx dy
= ∫(y = -4 to 4) xy^2 {for x = -2 to 2} dy
= ∫(y = -4 to 4) 4y^2 dy
= (4/3)y^3 {for y = -4 to 4}
= 512/3.

2) ∫(y = 1 to e) ∫(x = 0 to y) (x + y)^(-1) dx dy
= ∫(y = 1 to e) ln(x + y) {for x = 0 to y} dy
= ∫(y = 1 to e) (ln(2y) - ln y) dy
= ∫(y = 1 to e) ln(2y/y) dy
= ∫(y = 1 to e) ln 2 dy
= (e - 1) ln 2.

3) ∫(x = 0 to 1) ∫(y = x to x+1) e^x dy dx
= ∫(x = 0 to 1) ye^x {for y = x to x+1} dx
= ∫(x = 0 to 1) e^x dx
= e^x {for x = 0 to 1}
= e - 1.

I hope this helps!

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