let f be a continuous function such that ∫₀¹⁷ f(x)dx = 8, ∫₁⁷²⁰ f(x)dx = -3, and ∫₁₃²⁰ f(x)dx = 7. what is…

let f be a continuous function such that ∫₀¹⁷ f(x)dx = 8, ∫₁⁷²⁰ f(x)dx = -3, and ∫₁₃²⁰ f(x)dx = 7. what is the value of ∫₀¹³ f(x)dx? a -2 b 4 c 12 d 18

let f be a continuous function such that ∫₀¹⁷ f(x)dx = 8, ∫₁⁷²⁰ f(x)dx = -3, and ∫₁₃²⁰ f(x)dx = 7. what is the value of ∫₀¹³ f(x)dx? a -2 b 4 c 12 d 18

Answer

Explanation:

Step1: Use integral property

We know that $\int_{a}^{c}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx$. So, $\int_{0}^{20}f(x)dx=\int_{0}^{17}f(x)dx+\int_{17}^{20}f(x)dx$. Given $\int_{0}^{17}f(x)dx = 8$ and $\int_{17}^{20}f(x)dx=- 3$, then $\int_{0}^{20}f(x)dx=8+( - 3)=5$.

Step2: Use integral property again

Also, $\int_{0}^{20}f(x)dx=\int_{0}^{13}f(x)dx+\int_{13}^{20}f(x)dx$. We know $\int_{0}^{20}f(x)dx = 5$ and $\int_{13}^{20}f(x)dx = 7$. Rearranging for $\int_{0}^{13}f(x)dx$, we get $\int_{0}^{13}f(x)dx=\int_{0}^{20}f(x)dx-\int_{13}^{20}f(x)dx$.

Step3: Calculate the result

Substitute the values: $\int_{0}^{13}f(x)dx=5 - 7=-2$.

Answer:

A. -2