Finding the volume under graph

Question

Find the volume under the graph of f between the planes x=a,x=b,y=c,y=d. f(x,y)=x³+y²+2 ; a=-1,b=1,c=1,d=3.

Answer 1

OK. Volume is a double integral: one integration on y and one on x.

 

V = ∫∫ x³+y²+2 dy dx (x from a to b, y from c to d)

 

 = ∫ x³ + (y³/3) + 2y dx ] y from c to d

 

 = ∫ x³ + (d³ - c³)/3 + 2(d - c) dx

 

 = (x⁴/4) + [(d³ - c³)/3 + 2(d - c)]x  (x from a to b)

 

 = (b⁴ - a⁴)/4 + [(d³ - c³)/3 + 2(d - c)](b - a)

Cheers!