Consider the integral I $=Z20Z212x2\sqrt[\phantom{2}]{y}cosydydx.i.$ Sketch the region $of$ integration. $ii.$ Set $up$ a double integral $to$ represent the area $of$ the region and calculate this area. iii. Reverse the order $of$ integration $in$ I and hence $or$ otherwise evaluate the double integral.

$2012$ Exam Questions

Question $1$

$(a)$ Consider the integral

$I={\displaystyle \underset{0}{\overset{2}{}}}{\displaystyle \underset{\frac{1}{2}{x}^2}{\overset{2}{}}}\sqrt[\phantom{2}]{y}cosydydx$

$i.$ Sketch the region $of$ integration.

$ii.$ Set $up$ a double integral $to$ represent the area $of$ the region and calculate this area.

iii. Reverse the order $of$ integration $in$ I and hence $or$ otherwise evaluate the double integral.

$(b)$ Consider the solid bounded $by$ the paraboloid $z=9-x^2-y^2$ and the $xy-$plane.

$i.$ Set $up$ a triple integral $to$ represent the volume $of$ this solid.

$ii.$ Evaluate the volume $of$ this solid.