Integrate f(x) = x from 0 to 4.

Step-by-step solution

1. 1

   Recognise which rule applies. The integrand $x$ is a power $x^{1}$, so the power rule $\int x^{n}dx=\dfrac{x^{n+1}}{n+1}+C$ with $n=1$ handles it — no substitution or parts needed.

2. 2

   Write down the antiderivative.

   $\int x\,dx=\frac{x^{2}}{2}+C$

   For a definite integral the constant $C$ can be dropped: it appears at both limits and cancels in the subtraction.

3. 3

   Apply the Fundamental Theorem of Calculus. Evaluate the antiderivative at the upper limit and subtract its value at the lower limit:

   $\int_{0}^{4}x\,dx=\left[\frac{x^{2}}{2}\right]_{0}^{4}=\frac{4^{2}}{2}-\frac{0^{2}}{2}=\frac{16}{2}-0=8$

4. 4

   Check the result geometrically. On $[0,4]$ the graph of $y=x$ is a straight line, so the region under it is a right triangle with base $4$ and height $4$:

   $\text{Area}=\frac{1}{2}\cdot 4\cdot 4=8$

   The geometric area matches the antiderivative computation exactly, which is the strongest available confirmation for a linear integrand.

5. 5

   State the answer.

   $\int_{0}^{4}x\,dx=8$