Question

You are given the following pieces of information:

₃₃∫⁸f(x)dx=−52, ₃₃∫¹² f(x)dx=−23, ₀∫ ¹³f(x)dx=64.
What is the value of ₁₂∫¹³f(x)dx ?

Answer 1

Using properties of definite integrals, we combined the given integrals to find ₁₂∫₁₃f(x)dx which turned out to be 0.

Step-by-step explanation:

To find the value of the integral ₁₂∫₁₃f(x)dx, we need to use the properties of definite integrals and the given information. We have:

• ₃₃∫₈f(x)dx = −52
• ₃₃∫₁₂f(x)dx = −23
• ₀∫₁₃f(x)dx = 64

The third integral, ₀∫₁₃f(x)dx, can be broken down into:

₀∫₁₃f(x)dx = ₀∫₃f(x)dx + ₃∫₈f(x)dx + ₈∫₁₂f(x)dx + ₁₂∫₁₃f(x)dx

Since we know the values of the integrals from 3 to 8 and from 3 to 12, we can find the first two parts of this expression:

₀∫₃f(x)dx = ₀∫₁₃f(x)dx - ₃₃∫₈f(x)dx - ₃₃∫₁₂f(x)dx

₀∫₃f(x)dx = 64 - (−52) - (−23) = 64 + 52 + 23

₀∫₃f(x)dx = 139

Finally, to get ₁₂∫₁₃f(x)dx, we subtract all the known parts from the total:

₁₂∫₁₃f(x)dx = ₀∫₁₃f(x)dx - ₀∫₃f(x)dx - ₃₃∫₈f(x)dx - ₈∫₁₂f(x)dx

₁₂∫₁₃f(x)dx = 64 - 139 - (−52) - (−23) = 64 - 139 + 52 + 23

₁₂∫₁₃f(x)dx = 0