4 [*ƒ 2 [ (8f(x) 11g(x)) dx = Use the fact that [²ƒ(2) dz = f(z) = [₁ (81(x) - f(x) dx = 26 and 4 Lª 9(2 g(x) dx = 24 to compute each of the following:
4 [*ƒ 2 [ (8f(x) 11g(x)) dx = Use the fact that [²ƒ(2) dz = f(z) = [₁ (81(x) - f(x) dx = 26 and 4 Lª 9(2 g(x) dx = 24 to compute each of the following:
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![**Computing Definite Integrals Using Given Values**
Given the fact that:
\[ \int_{2}^{4} f(x) \, dx = 26 \]
and
\[ \int_{2}^{4} g(x) \, dx = -24 \]
we can compute each of the following:
1. \[ \int_{4}^{2} f(z) \, dz \]
2. \[ \int_{2}^{4} (8f(x) - 11g(x)) \, dx \]
### Solution:
1. **Evaluating \(\int_{4}^{2} f(z) \, dz \)**:
Note that reversing the limits of integration changes the sign of the integral:
\[
\int_{4}^{2} f(z) \, dz = -\int_{2}^{4} f(z) \, dz = -26
\]
2. **Evaluating \(\int_{2}^{4} (8f(x) - 11g(x)) \, dx \)**:
Use the properties of integrals to separate and scale the given integrals:
\[
\int_{2}^{4} (8f(x) - 11g(x)) \, dx = 8 \int_{2}^{4} f(x) \, dx - 11 \int_{2}^{4} g(x) \, dx
\]
Substitute the given values:
\[
8 \int_{2}^{4} f(x) \, dx - 11 \int_{2}^{4} g(x) \, dx = 8 \cdot 26 - 11 \cdot (-24)
\]
Calculate:
\[
8 \cdot 26 = 208
\]
\[
-11 \cdot (-24) = 264
\]
Therefore:
\[
208 + 264 = 472
\]
So, the computed integrals are:
1. \[ \int_{4}^{2} f(z) \, dz = -26 \]
2. \[ \int_{2}^{4} (8f(x) - 11g(x)) \, dx = 472 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7f31f885-3acd-4362-8d52-f6c21adcaba6%2Fdcb5c65e-c074-4514-8507-3849421a67e9%2F9x745m_processed.png&w=3840&q=75)
Transcribed Image Text:**Computing Definite Integrals Using Given Values**
Given the fact that:
\[ \int_{2}^{4} f(x) \, dx = 26 \]
and
\[ \int_{2}^{4} g(x) \, dx = -24 \]
we can compute each of the following:
1. \[ \int_{4}^{2} f(z) \, dz \]
2. \[ \int_{2}^{4} (8f(x) - 11g(x)) \, dx \]
### Solution:
1. **Evaluating \(\int_{4}^{2} f(z) \, dz \)**:
Note that reversing the limits of integration changes the sign of the integral:
\[
\int_{4}^{2} f(z) \, dz = -\int_{2}^{4} f(z) \, dz = -26
\]
2. **Evaluating \(\int_{2}^{4} (8f(x) - 11g(x)) \, dx \)**:
Use the properties of integrals to separate and scale the given integrals:
\[
\int_{2}^{4} (8f(x) - 11g(x)) \, dx = 8 \int_{2}^{4} f(x) \, dx - 11 \int_{2}^{4} g(x) \, dx
\]
Substitute the given values:
\[
8 \int_{2}^{4} f(x) \, dx - 11 \int_{2}^{4} g(x) \, dx = 8 \cdot 26 - 11 \cdot (-24)
\]
Calculate:
\[
8 \cdot 26 = 208
\]
\[
-11 \cdot (-24) = 264
\]
Therefore:
\[
208 + 264 = 472
\]
So, the computed integrals are:
1. \[ \int_{4}^{2} f(z) \, dz = -26 \]
2. \[ \int_{2}^{4} (8f(x) - 11g(x)) \, dx = 472 \]
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