The volume determined from ∫∫∫v 8 xyz dv for V = [2, 3] × [1, 2] × [ 0,1 ] will be (in integer) ________.
Explanation
Given
Integral
∫∫∫v 8 xyz dv
Limits for x, y and z is given as
[2, 3] × [1, 2] × [0, 1]
Volume of the integral
V = ∫∫∫v 8 xyz dv
i.e. V = ∫ ∫ ∫V 8 xyz dxdydz
\(V = 8 \times \mathop \smallint \limits_2^3 xdx\mathop \smallint \limits_1^2 ydy\mathop \smallint \limits_0^1 zdz\)
\(V = 8 \times \left[ {\frac{{{x^2}}}{2}} \right]_2^3 \times \left[ {\frac{{{y^2}}}{2}} \right]_1^2 \times \left[ {\frac{{{z^2}}}{2}} \right]_0^1\)
V = 5 × 3 × 1
V = 15
∴ Volume is 15
Important Questions from Vector Calculus
The points with position vectors 60î + 3ĵ, 40î -8ĵ, aî - 52ĵ are collinear if a is equal to
If A = 3i + j + k; B = 5i + j – k; C = i + j - k then find the volume of parallelogram if A, B, and C are the sides of the parallelepiped respectively.
If f(x, y) = 0 then find the directional derivative at c = (0, 0) along the direction u = (a, b)?
Find the value of \(\int \int Curl \vec F. d\vec r\) where F(x, y, z) = (y + z, z + x, x + y)
The functions which are present on one side of Green's theorem are of which kind?
