Consider the function:

Question

f(x)= {−2, if1≤x<7

4, if7≤x≤13

Evaluate the following definite integral using properties of definite integrals.

integral from 1 to 13 of f(x)dx=

Answer 1

We can break up the interval of integration based on the piecewise function definition. Then within each interval, we use the corresponding function definition and evaluate as usual.

∫₁¹³f(x)dx

= ∫₁⁷f(x)dx + ∫₇¹³f(x)dx

= ∫₁⁷ -2 dx + ∫₇¹³ 4 dx

= -2x |₁⁷ + 4x |₇¹³

= -2*7 - (-2)*1 + 4*13 - 4*7

= -14 + 2 + 52 - 28

= 12

Answer 2