Calculus II, Spring 2024 (http://www.math.nthu.edu.tw/˜ wangwc/) Thomas’ Calculus Early Transcendentals 13ed

Study guide for quiz 11

Quiz problems include both the lecture contents and homework problems.

1. Section 15.3 (Area by double integral) is a special case of Section 15.2 (Double integrals in general domain) and will not appear in quiz 11.

2. Section 15.4:

Study why dA = rdrdθ in polar coordinates. Practice how to determine the lim-

its of integration in ∫_{θ_{1}}^{θ_{2}} ∫_{f_{1}(θ)}^{f_{2}(θ)}(⋯)rdrdθ as in Examples 2-6 of section 15.4. More

specifically:

(a) Given a domain R in the x − y plane, practice drawing θ = C lines in R. The end points of these lines are lower limit (the near end point) and upper limit (the far end point) of the integration rdr. The end points for dθ are smallest and largest C among these θ = C lines.

(b) The end points of the lines θ = C (that is, the lower limit and upper limit of rdr) must be expressed as r = f_{1}(θ) and r = f_{2}(θ). Given a simple curve F (x, y) = 0 (such as a line or a circle), use the substitution x = r cos θ, y = r cos θ to express it as r = f(θ). Examples: x = 1, y = −3, x + y = 1, x^{2} + y^{2} = 4, etc.

3. Section 15.5:

Practice how to determine the limits of integration for triple integrals in rectangular coordinates. For example, which cross section ( {x = constant}, {y = constant} or {x = constant}) is needed for dxdydz? which cross section is needed for dzdxdy? etc.

On a cross section, the triple integral reduces to double integral for the first two integration variables.

Note that, the upper and lower limits of the first (inner) variable may depend on the second (middle) and third (outer) variables. The upper and lower limits of the second variable may depend on the third variable.

Practice this on corresponding examples and exercises in section 15.5.