Question

evaluate the integral
10
6
6 dx.
10
6
6 dx=
(simplify your answer.)

Explanation:

Step1: Apply the integral formula

The integral of a constant \(k\) with respect to \(x\) is \(kx + C\). For \(\int 6dx\), the antiderivative is \(6x\).

Step2: Use the fundamental theorem of calculus

The fundamental theorem of calculus states that \(\int_{a}^{b}f(x)dx=F(b)-F(a)\), where \(F(x)\) is the antiderivative of \(f(x)\). Here \(a = 6\), \(b = 10\), and \(F(x)=6x\). So \(\int_{6}^{10}6dx=6x\big|_{6}^{10}\).

Step3: Calculate \(F(b)-F(a)\)

Substitute \(x = 10\) and \(x = 6\) into \(6x\). We get \(6\times10-6\times6\).

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Answer:

\(24\)