The area under the curve y = x⁴ and the lines x = 1, x = 5 and x-axis is:

Concept:

The area under the function y = f(x) from x = a to x = b and the x-axis is given by the definite integral 

​\(\rm \displaystyle \left|\int_a^b f(x)\ dx\right|\)

This is for curves that are entirely on the same side of the x-axis in the given range.

If the curves are on both sides of the x-axis, then we calculate the areas of both sides separately and add them.

Definite integral: If ∫ f(x) dx = g(x) + C, then \(\rm \displaystyle \int_a^b f(x)\ dx = [ g(x)]_a^b=g(b)-g(a).\)

\(\rm \displaystyle \int x^n\ dx = \dfrac{x^{n+1}}{n+1}+C\).

Calculation:

\(\rm \displaystyle \int x^4\ dx = \dfrac{x^5}{5}+C\).

Using the above concept for area of a curve, we can say that the required area is:

\(\rm I=\displaystyle \int_1^5 x^4\ dx\)

\(\rm = \left [\dfrac{x^5}{5}\right ]_1^5\)

\(\rm =\dfrac{5^5}{5}-\dfrac{1^5}{5}\)

\(\rm =\dfrac{3125-1}{5}\)

\(\rm =\dfrac{3124}{5}\).