- #1

karush

Gold Member

MHB

- 3,269

- 5

b. Evaluate

$V=5\displaystyle\int_0^8 \biggr[ x^3\biggr]_{(y-4)/2}^{y^{1/3}}\ dy \

=5\displaystyle\int_0^8 [(y^{1/3})^3-((y-4)/2))^3] \quad \ dy \

=5\displaystyle\int_0^8 \biggr [y-\dfrac{(y-4)^3}{8}\biggr] \ dy$

Expand $5\displaystyle\int_0^8 -\dfrac{y^3}{8}+\dfrac{3y^3}{2}-5y +8 \ dy

=5\biggr[-\dfrac{y^4}{32}+\dfrac{y^3}{2}-\dfrac{5y^2}{2}+8y \biggr]_0^8

=\biggr[-\dfrac{8^4}{32}+\dfrac{8^3}{2}-\dfrac{5(8)^2}{2}+8(8) \biggr]=160$

ok there is bound to be some typos in this or too much content

want to make sure this is correct before i attempt a tikz of the region

mahalo for suggestions and such