Math 9B Worksheet 9 (M) Fall 2023
This WS covers the second part of section 6.8 and section 7.1.
1. (4 points) Determine if the integrals are convergent of divergent
(a) (Use one of: Integration by parts or direct comparison test)
∫ ∞
1
ln(x)
x4
dx
(b) (Use limit comparison test, or direct comparison test)
∫ ∞
3
1
x+ ex
dx
2. (4 points) Let R be the finite region bounded by given functions. In the following way,
set up and evaluate definite integrals which represent the area of the region R: y = e4x,
y = 81, and x = 0.
(a) Sketch a graph of the region R. Label all intersection points.
(b) Integrate with respect to x.
(c) Integrate with respect to y.
(d) Explain the basic idea behind the formula for area between two curves.
3. (2 points) Sketch the region bound by the curves y =
3
x2
, y = 3x, and the x-axis, labelling
all intersection points and find the area.
4. (optional extra practice) You don’t need to turn these in.
ˆ Find the area of the region between the
curves y = sin
(πx
2
)
and y = x.
ˆ Find the area of the region between the
curves y = 20/x and y = −2x+ 14
ˆ Find the area of the region between the
curves y = 2x3 and y = 5x2
ˆ Find area of the regionR : 0 ≤ y ≤ e−
√
x
on the interval 0 ≤ x ≤ ∞.