Question

1. find the area under the graph of y = x³ between x = 0 and x = 2.

Explanation:

Step1: Use the definite integral formula

The area \(A\) under the curve \(y = f(x)\) from \(x = a\) to \(x = b\) is given by \(A=\int_{a}^{b}f(x)dx\). Here, \(f(x)=x^{3}\), \(a = 0\), and \(b = 2\). So, \(A=\int_{0}^{2}x^{3}dx\).

Step2: Apply the power - rule for integration

The power - rule for integration is \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)). For \(n = 3\), \(\int x^{3}dx=\frac{x^{4}}{4}+C\).

Step3: Evaluate the definite integral

Using the fundamental theorem of calculus \(\int_{a}^{b}F^{\prime}(x)dx=F(b)-F(a)\). If \(F(x)=\frac{x^{4}}{4}\), then \(F(2)-F(0)=\frac{2^{4}}{4}-\frac{0^{4}}{4}\).

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Answer:

\(4\)