Download MA 114: Calculus II Exam 3 Review (New Material) §8.1 Arc Length

MA 114: Calculus II
Exam 3 Review (New Material)
§8.1 Arc Length and Surface Area
formula to calculate arc length of a curve; formula to calculate surface area of a solid of
revolution
§8.3 Center of Mass
definition of a moment; formula to calculate My (moment with respect to y-axis); two formulas
to calculate Mx (moment with respect to x-axis; how to find the center of mass of a lamina
§9.1 Solving Differential Equations
order of a differential equation; linearity of a differential equation; how to solve a separable
differential equation
§9.2 Models Involving y 0 = k(y − b)
set up and solve the differential equation corresponding to Newton’s Law of Cooling and
free-fall with air resistance; focus on word problems
§9.3 Graphical and Numerical Methods
how to construct/identify the slope field corresponding to a given differential equation; how
to plot integral curves
§9.4 The Logistic Equation
set up and solve the differential equation corresponding to a relevant word problem
§9.5 First-Order Linear Differential Equations
be able to identify a first-order linear differential equation, find the integration factor, and
find the general and/or particular solution
§11.1 Parametric Equations
how to parametrize basic functions; find the derivative at a point
§11.2 Arc Length and Speed (from parametric equations)
formula to calculate arc length of a parametric curve; formula to calculate speed at a point on
a parametric curve; formula to calculate surface area of solid of revolution of a parametrization
§11.3 Polar Coordinates
how to convert between rectangular and angular coordinates; convert between a polar function
and a rectangular function
§11.4 Area and Arc Length in Polar Coordinates
formula to calculate area based on polar coordinates; formula to calculate arc length of a polar
curve
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1. Set up (but do not evaluate) integrals for the following geometric quantities:
a. The length of the curve y = x2 from x = a to x = b.
b. The surface area of the solid obtained by rotating the region bounded by the curves
y = 1/x, x = 1, and x = 5 about the x-axis.
c. The length of the parametric curve x = 3t2 , y = 2t3 on 0 ≤ t ≤ 2.
d. The area enclosed by the polar curve r = 1 − cos θ.
2. Consider the curve with parametric equations x(t) = 3t2 + t and y(t) = 2t.
a.) Eliminate the parameter t to find a Cartesian equation for this curve.
b.) Find the tangent line to this curve at the point (x, y) = (14, 4).
3. Consider the polar curves r = sin(2θ) and r = cos θ.
a.) Determine the Cartesian coordinates (x, y) of the point of intersection which is strictly
in the first quadrant, that is, when x > 0 and y > 0.
b.) Set up the integrals for computing the areas of each of the three regions in the first
quadrant bounded by these two functions and the x-axis. Do not evaluate the integrals.
4. Solve the initial-value problem
dL
= kL2 ln t, L(1) = −1.
dt
5. Draw the direction field for the differential equation y 0 = y + x. Sketch the solution which
satisfies y(0) = 0.
6. Find the solution to y 0 − y = e2x with y(0) = 1.
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