Question

1. Let R be the rectangle in the xy-plane bounded by the lines x = 1, x = 4, y = −1, and y = 2. Evaluate Z Z R sin(πx + πy) dA.

2. Let T be the triangle with vertices (0, 0), (0, 2), and (1, 0). Evaluate the integral Z Z T xy^2 dA

ZZ means double integral. All x's are variables. Thank you!.

Answer #1

2. Evaluate the double integral Z Z R e ^(x^ 2+y ^2) dA where R
is the semicircular region bounded by x ≥ 0 and x^2 + y^2 ≤ 4.
3. Find the volume of the region that is bounded above by the
sphere x^2 + y^2 + z^2 = 2 and below by the paraboloid z = x^2 +
y^2 .
4. Evaluate the integral Z Z R (12x^ 2 )(y^3) dA, where R is the
triangle with vertices...

Let f(x,y)=x2ex2f(x,y)=x2ex2 and let RR be the
triangle bounded by the lines x=2x=2, x=y/3x=y/3, and y=xy=x in the
xyxy-plane.
(a) Express ∫RfdA∫RfdA as a double integral in
two different ways by filling in the values for the integrals
below. (For one of these it will be necessary to write the double
integral as a sum of two integrals, as indicated; for the other, it
can be written as a single integral.)
∫RfdA=∫ba∫dcf(x,y)d∫RfdA=∫ab∫cdf(x,y)d dd
where a=a= , b=b= , c=c= , and
d=d= .
And...

Consider the integral ∫∫R(x^2+sin(y))dA where R is the region
bounded by the curves x=y^2, x=4, and y=0. Setup up this
integral.

Use the given transformation to evaluate the integral.
6xy dA
R
, where R is the region in the first quadrant bounded
by the lines y =
1
2
x and y =
3
2
x and the hyperbolas xy =
1
2
and xy =
3
2
;
x = u/v, y = v

Let S be the boundary of the solid bounded by the paraboloid
z=x^2+y^2 and the plane z=16
S is the union of two surfaces. Let S1 be a portion of the plane
and S2 be a portion of the paraboloid so that S=S1∪S2
Evaluate the surface integral over S1
∬S1 z(x^2+y^2) dS=
Evaluate the surface integral over S2
∬S2 z(x^2+y^2) dS=
Therefore the surface integral over S is
∬S z(x^2+y^2) dS=

Let R be the region of the plane bounded by y=lnx and the x-axis
from x=1 to x= e. Draw picture for each
a) Set up, but do not evaluate or simplify, the definite
integral(s) that computes the volume of the solid obtained by
rotating the region R about they-axis using the disk/washer
method.
b) Set up, but do not evaluate or simplify, the definite
integral(s) that computes the volume of the solid obtained by
rotating the region R about...

Q8. Let G be the cylindrical solid bounded by x2 + y2 = 9, the
xy-plane, and the plane
∫∫
z = 2, and let S be its surface. Use the Divergence Theorem to
evaluate I = S F · ndS where F(x,y,z) = x3i + y3j + z3k and n is
the outer outward unit normal to S.

Use the given transformation to evaluate the integral.
6xy dA
R
, where R is the region in the first quadrant bounded
by the lines y =
2
3
x and y =
3
2
x and the hyperbolas xy =
2
3
and xy =
3
2
;
x = u/v, y = v

57.
a. Use polar coordinates to compute the (double integral (sub
R)?? x dA, R x2 + y2) where R is the region in the first quadrant
between the circles x2 + y2 = 1 and x2 + y2 = 2.
b. Set up but do not evaluate a double integral for the mass of
the lamina D={(x,y):1≤x≤3, 1≤y≤x3} with density function ρ(x, y) =
1 + x2 + y2.
c. Compute??? the (triple integral of ez/ydV), where E=
{(x,y,z):...

Consider the plane region R bounded by the curve y = x − x 2 and
the x-axis. Set up, but do not evaluate, an integral to find the
volume of the solid generated by rotating R about the line x =
−1

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