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By changing the order of integration, show that : Z x
0
Z u
0
e
m(x−t)
f(t) dt du =
Z x
0
(x − t) e
m(x−t)
f(t) dt.
Show, by changing to polar coordinates, that Z a sin β
Z
√
a2−y2
y cot β
ln (x
2 + y
2
) dx dy = a
2β(ln a − 1/2), where a > 0 and
0 < β < π/2. Rewrite the Cartesian integral with the order of integration reversed.
Find the average height of the paraboloid z = x
2 + y
2 over the square 0 ≤ x ≤ 2, 0 ≤ y ≤ 2.
What region R in the xy-plane maximizes the value of RR
R
(4−x
2 −2y
2
) dA? What region R in the xy-plane minimizes
the value of RR
R
(x
2 + y
2 − 9) dA? Give the reason for your answer.
Use a double integral in polar coordinates to find the area of the region common to the interior of the cardioids
r = 1 + cos θ and r = 1 − cos θ.
Evaluate the double integral as iterated integral in two ways : RR
R
xy2 dA; R is the region enclosed by y = 1, y = 2, x = 0,
and y = x
Let 𝑓(𝑥) = ൝ 1 + 2𝑥, 𝑥 ≤ 0 3𝑥 − 2,0 < 𝑥 ≤ 1 2𝑥 ଶ − 1, 𝑥 > 1 i) Check whether f is discontinuous. If yes, find where? ii) Give a rough sketch of the graph of f.
Determine the concavity , y-intercept , x-intercepts and co-ordinates of vertex of the parabola 𝑓(𝑥) = 5𝑥 2 − 𝑥 − 3.
Find the dimension of rectangular box with the largest possible volume with an open top and one portion to be constructed from 162 sq. inches of cardboard. (Note: The amount of the material used in construction of box is xy+2xz+2yz=162)
A stone is dropped into a pool of water. The radius of the circular ripple formed increases at 4 𝑚/𝑠. Calculate the rate at which the area of the ripple is increasing when the radius is 6m.
#340153 on Dec 2023