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# Exercise 10 (12 points). Let A C R3 be Jordan-measurable in R3. Dene the function a: R ) [0: DO) by 001): 52({(w,y) E 132 | With) 6 A}), that is, the...

Let A ⊂ R 3 be Jordan-measurable in R 3 . Define the function a: R → [0, ∞) by a(h) = λ 2 ({(x, y) ∈ R 2 | (x, y, h) ∈ A}), that is, the area of {(x, y) ∈ R 2 | (x, y, h) ∈ A}, for all h ∈ R. 1. Justify that a is well-defined. 2. Let h1 = inf{h ∈ R | a(h) > 0} and h2 = sup{h ∈ R | a(h) > 0}. Show that if the restriction of a to [h1, h2] is a polynomial of degree at most 3, then the volume of A is given by λ 3 (A) = h2 − h1 6 a(h1) + a(h2) + 4a h1 + h2 2 . 3. Using the above formula, find the volume of a cone of height H > 0 and base area B > 0.

Exercise 10 (12 points). Let A C R3 be Jordan-measurable in R3. Deﬁne the function a: R —)
[0: DO) by
001): 52({(w,y) E 132 | With) 6 A}), that is, the area of {(w,y) E R2 | (stay, h) E A}, for all h e R. 1. (2 points) Justify that a. is well-deﬁned.
2. (8 points) Let h1 = inf{h E R | (JUL) &gt; 0} and h2 = sup{h E R | a(h) &gt; 0}. Show that if the restriction of a to [h1, h2] is a polynomial of degree at most 3, then the
volume of A is given by AM) = M g’” (0:011) +a(h2) +4“ 013”». 3. (2 points) Using the above formula, ﬁnd the volume of a cone of height H &gt; 0 and base
area B &gt; 0.

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