Question

x, y, z ന്ടെ ലിമിറ്റ് V=[2,3] x [1,2] x [0, 1] ആണെങ്കിൽ $\int\int\int_v8xyzdv$ൽ നിന്നുമുള്ള വ്യാപ്തം ആണ്

• A. 15
• B. 50
• C. 25
• D. 51

Answer 1

Answer:

A. 15

Given:
$V = [2,3] \times [1,2] \times [0,1]$

and

$\iiint_V 8xyz , dV$

Split the integral

Since limits are independent,

$\iiint_V 8xyz , dV = 8 \left(\int_2^3 x,dx\right)\left(\int_1^2 y,dy\right)\left(\int_0^1 z,dz\right)$

Evaluate each

$\int_2^3 x,dx = \left[\frac{x^2}{2}\right]_2^3 = \frac{9}{2} - \frac{4}{2} = \frac{5}{2}$

$\int_1^2 y,dy = \left[\frac{y^2}{2}\right]_1^2 = \frac{4}{2} - \frac{1}{2} = \frac{3}{2}$

$\int_0^1 z,dz = \left[\frac{z^2}{2}\right]_0^1 = \frac{1}{2}$

Multiply

$8 \times \frac{5}{2} \times \frac{3}{2} \times \frac{1}{2}$

$= 8 \times \frac{15}{8} = 15$

Final Answer:

$\boxed{15}$