Questions: Consider the function f(x)= x if x<1 1 if 1 ≤ x ≤ 3 -4x+13 if x>3 Sketch a graph of y=f(x) use the graph to evaluate the definite integral ∫-2^5 f(x) dx

Solution Steps

For \(x < 1\), \(f(x) = x\). Therefore, the integral from \(-2\) to \(1\) is:

\[ \int_{-2}^{1} x \, dx = \left[ \frac{x^2}{2} \right]_{-2}^{1} = \frac{1^2}{2} - \frac{(-2)^2}{2} = \frac{1}{2} - 2 = -\frac{3}{2} \]

For \(1 \leq x \leq 3\), \(f(x) = 1\). Therefore, the integral from \(1\) to \(3\) is:

\[ \int_{1}^{3} 1 \, dx = \left[ x \right]_{1}^{3} = 3 - 1 = 2 \]

For \(x > 3\), \(f(x) = -4x + 13\). Therefore, the integral from \(3\) to \(5\) is:

\[ \int_{3}^{5} (-4x + 13) \, dx = \left[ -2x^2 + 13x \right]_{3}^{5} = \left( -2(5)^2 + 13(5) \right) - \left( -2(3)^2 + 13(3) \right) \]

\[ = \left( -50 + 65 \right) - \left( -18 + 39 \right) = 15 - 21 = -6 \]

Final Answer