Question

Question:To determine the exact value of the integral

Short answer

The exact value of the integralis \(\underline {0.12507} \).

Step-by-step solution

Concept of Surface Integral and Formula Used

A surface integral is a surface integration generalisation of multiple integrals. It can be thought of as the line integral's double integral equivalent. It is possible to integrate a scalar field or a vector field over a surface.

Definition used:

The surface area of surface \({\rm{S}}\) with the equation \(z = g(x,y)\) is given by the formula

Calculate the Surface Integral

The given equation of the surface is \(z = xy\) where \(0 \le x \le 1,0 \le y \le 1\)

Differentiate the given equation with respect to\(x\)and obtain the value of\(\frac{{\partial z}}{{\partial x}}\)as shown below:

\(\begin{array}{c}\frac{{\partial z}}{{\partial y}} = \frac{{\partial (xy)}}{{\partial x}}\\ = x\end{array}\)

Differentiate the given equation with respect to\(y\)and obtain the value of\(\frac{{\partial z}}{{\partial y}}\)as shown below:

\(\begin{array}{c}\frac{{\partial z}}{{\partial x}} = \frac{{\partial (xy)}}{{\partial x}}\\ = y\end{array}\)

Substitute \(\frac{{\partial z}}{{\partial y}} = x,\frac{{\partial z}}{{\partial x}} = y\) in obtain the exact value of the integral as follows:

Therefore,

Using the Wolfram Alpha calculator obtain the value of the above integral, it can be observed that the exact value of the integral:

\(\int_0^1 {\int_0^1 {{x^3}} } {y^2}\sqrt {1 + {{(x)}^2} + {{(y)}^2}} dxdy = 0.12507\)

Thus, the exact value of the integral is \(\underline {0.12507} \).