1. Create a double integral (with correct limits) that computes the volume bounded between
z =x2 + 10 and z = y2 10 where x 2 [-1,1] and y 2 [-1,1]. Solve it.
The double integral is:
The volume is:
2. Create a triple integral (with correct limits) that computes the volume bounded between
z = x2 y2 + 1 and the x y plane, in the positive orthont. Integrate in the order dz, dy, dx. Solve it,
showing key steps (the boxes) below.
The triple integral is:
After simpli cation, the double integral is:
After simpli cation, the single integral is:
The volume is:
c S.J. Gismondi (Instructor), 2015.
All rights reserved.
3. Construct the triple integral, in the order dx, dy, dz that computes the nite volume in the
positive orthont bounded by x + 2y + 3z = 6. Complete the boxes below and compute the volume.
The triple integral is:
After simpli cation, the double integral is:
After simpli cation, the single integral is:
The volume is:
4. From above, construct the triple integral again but in the order dz, dy, dx and repeat the volume computation.
Complete the boxes below.
The triple integral is:
After simpli cation, the double integral is:
After simpli cation, the single integral is:
The volume is:
5. Consider the volume of the region bounded above by z = x2 + y2 + 1 and the x ?? y plane, where x 2 [0,1] and y 2 [0,1]. Do the following.
a) Draw a picture of the volume such that the x axis is partitioned into four equal parts and the y axis
is partitioned into two equal parts. These will be called subregions in the xy plane. Be sure to clearly label x, y and z axes.
b) Explicitly construct/write the sum of the volumes of these eight rectangular boxes, each box having a base de ned by a
subregion, and where z i is the height of each box on each subregion (Compute z i in the very middle of each subregion.). This
is call the Riemann sum. Show your work here.
c) Compute the numerical value of the Riemann sum. Show your work here.