Question

Evaluate ∬ D ( x + y ) d A

where D is the region
defined by the parabola y = -x^{2} + 4, y=0(x-axis) and
x=0(y-axis)

Answer #1

Evaluate the double integral of 5x3cos(y3)
dA where D is the region bounded by y=2, y=(1/4)x2, and
the y-axis.

Evaluate ∬D15x2−6ydA where D is the region bounded by
x=1/2y^2and x=4√y

Evaluate ∫∫R(6xy+4)dA, ∫ ∫ R ( 6 x y + 4 ) d A , where R R is
the region bounded by y=x2 y = x 2 and y=x+2 y = x + 2 . (Round
your answer to 2 decimal places)

evaluate the double integral where f(x,y) = 6x^3*y - 4y^2 and D
is the region bounded by the curve y = -x^2 and the line x + y =
-2

Evaluate the triple integral _ D sqrt(x^2+y^2+z^2) dV, where D
is the solid region given by 1 (less than or equal to) x^2+y^2+z^2
(less than or equal to) 4.

Given that D is a region bounded by x = 0, y = 2x, and y =
2.
Given:
∫
∫
x y dA , where D is the region bounded by x = 0, y = 2x, and y =
2.
D
Set up iterated integrals (2 sets) for both orders of
integration. Need not evaluate the
Integrals. Hint: Draw a graph of the region D. Consider D as a
Type 1 or Type 2 region.
Extra credit problem

Let D be the solid region defined by D = {(x, y, z) ∈ R3; y^2 +
z^2 + x^2 <= 1},
and V be the vector field in R3 defined by: V(x, y, z) = (y^2z +
2z^2y)i + (x^3 − 5^z)j + (z^3 + z) k.
1. Find I = (Triple integral) (3z^2 + 1)dxdydz.
2. Calculate double integral V · ndS, where n is pointing
outward the border surface of V .

A. For the region bounded by y = 4 − x2 and the x-axis, find
the volume of solid of revolution when the area is revolved
about:
(I) the x-axis,
(ii) the y-axis,
(iii) the line y = 4,
(iv) the line 3x + 2y − 10 = 0.
Use Second Theorem of Pappus.
B. Locate the centroid of the area of the region bounded by y
= 4 − x2 and the x-axis.

The average value of a function f(x, y, z) over a solid region E
is defined to be fave = 1 V(E) E f(x, y, z) dV where V(E) is the
volume of E. For instance, if ρ is a density function, then ρave is
the average density of E. Find the average value of the function
f(x, y, z) = 5x2z + 5y2z over the region enclosed by the paraboloid
z = 4 − x2 − y2 and the...

Evaluate the following integrals:
a.) Find the area enclosed by y = (ln(x))/ (x^2) and y =
(ln(x))^2/x2 ;
b.) Find the volume of the solid formed by revolving the region
under y = e^ 3x for 0 ≤ x ≤ 3 about the y -axis.

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