Title: Math 2260 Chapter 6 (Volume, Length, Area, Work)

Q: The volume of the portion of solid of cross-sectional area [$]A(x)[/$] with [$]a \leq x \leq b[/$] is
A: [$]\operatorname{Vol} = \int_a^b A(x) \, dx.[/$]

Q: The volume of the portion of the solid created by rotating [$]y = f(x)[/$] around the [$]x[/$]-axis (with [$]a \leq x \leq b[/$]) is computed by the disk method using the integral
A: [$]\operatorname{Vol} = \int_a^b \pi [R(x)]^2 \, dx = \int_a^b \pi f(x)^2 \, dx.[/$]

Q: The volume of a solid of revolution with inner radius [$]r(x)[/$] and outer radius [$]R(x)[/$] (rotated around the [$]x[/$] axis) is computed using the washer method by the integral
A: [$]\operatorname{Vol} = \int_a^b \pi[R(x)]^2 - \pi [r(x)]^2 \, dx.[/$]

Q: The volume of a region can be computed by the shell method using the integral
A: [$]\operatorname{Vol} = \int 2 \pi (\text{radius of shell})(\text{height of shell}) \, dr[/$]

Q: The length of the portion of the graph of [$]y = f(x)[/$] for [$]a \leq x \leq b[/$] is
A: [$]\operatorname{Length} = \int_a^b \sqrt{1 + f'(x)^2} \, dx.[/$]

Q: The length of the portion of the parametrized curve [$](x(t),y(t))[/$] with [$]a \leq t \leq b[/$] is
A: [$]\operatorname{Length} = \int_a^b \sqrt{ x'(t)^2 + y'(t)^2} [/$]

Q: The area of portion of the surface of revolution swept out by the graph [$]y = f(x)[/$] for [$]a \leq x \leq b[/$] as it is revolved around the x axis is given by
A: [$]\operatorname{Area} = \int_a^b 2 \pi f(x) \sqrt{ 1 + f'(x)^2 } \, dx.[/$]

Q: The work done by a variable force [$]F(x)[/$] in the direction of motion along the [$]x[/$] axis is given by
A: [$]\operatorname{Work} = \int F(x) \, dx[/$]

Q: Hooke's law for springs. The force required to hold a spring [$]x[/$] units from its natural (unstretched or uncompressed) length is
A: [$]k x[/$], where [$]k[/$] is called the spring constant

Q: Let [$]x[/$] be the depth below a pump. A tank under the pump has cross-sectional area [$]A(x)[/$], and contains fluid in the region [$]a \leq x \leq b[/$].
How much work is required to pump out the tank?
A: [$]\int_a^b x A(x) (\text{density}) g \, dx[/$]
if density is given in units of mass (eg. grams, kilograms), and [$]\int_a^b x A(x) (\text{density}) \, dx[/$]
if density is given in units of weight  (eg. lbs).