Questions: Suppose the integral from 0 to 4 of f(x) dx equals 3, the integral from 4 to 6 of f(x) dx equals -6, and the integral from 4 to 6 of g(x) dx equals 7. Evaluate the integrals in parts a - d. The integral from 0 to 4 of 5f(x) dx equals 15 The integral from 4 to 6 of -9g(x) dx equals -63 The integral from 4 to 6 of [8f(x)-g(x)] dx equals

Suppose the integral from 0 to 4 of f(x) dx equals 3, the integral from 4 to 6 of f(x) dx equals -6, and the integral from 4 to 6 of g(x) dx equals 7. Evaluate the integrals in parts a - d.
The integral from 0 to 4 of 5f(x) dx equals 15
The integral from 4 to 6 of -9g(x) dx equals -63
The integral from 4 to 6 of [8f(x)-g(x)] dx equals

Transcript text: Suppose $\int_{0}^{4} f(x) d x=3, \int_{4}^{6} f(x) d x=-6$, and $\int_{4}^{6} g(x) d x=7$. Evaluate the integrals in parts a - d. $\int_{0}^{4} 5 f(x) d x=15$ $\int_{4}^{6}-9 g(x) d x=-63$ $\int_{4}^{6}[8 f(x)-g(x)] d x=\square$

Solution

Solution Steps

To solve the given integrals, we will use the properties of definite integrals, such as linearity, which allows us to factor out constants and combine integrals.

For the first integral, use the property that allows you to factor out constants from the integral.
For the second integral, similarly, factor out the constant from the integral.
For the third integral, use the linearity of integrals to separate the integral into two parts and then apply the given values.

Step 1: Evaluate $ \int_{0}^{4} 5 f(x) \, dx $

Using the property of linearity of integrals, we can factor out the constant: \[ \int_{0}^{4} 5 f(x) \, dx = 5 \int_{0}^{4} f(x) \, dx = 5 \cdot 3 = 15 \]

Step 2: Evaluate $ \int_{4}^{6} -9 g(x) \, dx $

Again, applying the linearity of integrals: \[ \int_{4}^{6} -9 g(x) \, dx = -9 \int_{4}^{6} g(x) \, dx = -9 \cdot 7 = -63 \]

Step 3: Evaluate $ \int_{4}^{6} [8 f(x) - g(x)] \, dx $

We can separate the integral into two parts: \[ \int_{4}^{6} [8 f(x) - g(x)] \, dx = \int_{4}^{6} 8 f(x) \, dx - \int_{4}^{6} g(x) \, dx \] Calculating each part: \[ \int_{4}^{6} 8 f(x) \, dx = 8 \int_{4}^{6} f(x) \, dx = 8 \cdot (-6) = -48 \] Thus, \[ \int_{4}^{6} [8 f(x) - g(x)] \, dx = -48 - 7 = -55 \]