area x^2,0,0,1 — step-by-step solution

1) The area between

x^{2}

and

0

[0, 1]

is the integral of the absolute value of their difference

No interior intersection: the difference keeps its sign on the interval

[0, 1]

x^{2}

is above

0

: integrate

x^{2}

⇥1.1) Let's integrate:

\int_{0}^{1} x^{2} d x

⇥ⓘ An antiderivative of x^n is x^(n+1)/(n+1), for n ≠ -1:

\int x^{2} d x

\frac{x^{3}}{3}

▷

\frac{x^{3}}{3}

⇥1.2) Let's evaluate over the interval

[0, 1]

\frac{1^{3}}{3} - \frac{0^{3}}{3}

⇥

\frac{1}{3} + 0

⇥ⓘ Adding 0 changes nothing:

\frac{1}{3} + 0

\frac{1}{3}

⇥

\frac{1}{3}

The area on

[0, 1]

\frac{1}{3}

▶Area:

\frac{1}{3}

2) The numeric value of:

\frac{1}{3}

equals:

▷

\approx 0.3333