Question

b. $\int_{3}^{7}(f(x)-g(x)) d x=\square$ (simplify your answer.)
c. $\int_{3}^{5}(f(x)-g(x)) d x=\square$ (simplify your answer.)
d. $\int_{5}^{7}(g(x)-f(x)) d x=\square$ (simplify your answer.)
e. $\int_{5}^{7} 7 g(x) d x=\square$ (simplify your answer.)
f. $\int_{5}^{3} 4 f(x) d x=\square$ (simplify your answer.)

Explanation:

To solve these integral problems, we need to recall the properties of definite integrals, such as the difference rule \(\int_{a}^{b}(f(x) - g(x))dx=\int_{a}^{b}f(x)dx-\int_{a}^{b}g(x)dx\), the constant multiple rule \(\int_{a}^{b}k\cdot f(x)dx = k\cdot\int_{a}^{b}f(x)dx\) (where \(k\) is a constant), and the property of reversing the limits of integration \(\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx\). However, since the problem does not provide the specific functions \(f(x)\) and \(g(x)\) or their integral values over the given intervals, we cannot compute the numerical answers. We need more information (such as the graphs of \(f(x)\) and \(g(x)\), the values of \(\int_{a}^{b}f(x)dx\) and \(\int_{a}^{b}g(x)dx\) over the intervals \([3,5]\), \([5,7]\), etc.) to proceed with the calculations.

If we assume that we have the values of \(\int_{3}^{5}f(x)dx = F_1\), \(\int_{3}^{5}g(x)dx=G_1\), \(\int_{5}^{7}f(x)dx = F_2\), \(\int_{5}^{7}g(x)dx=G_2\), \(\int_{3}^{7}f(x)dx=F_1 + F_2\) and \(\int_{3}^{7}g(x)dx=G_1+G_2\), we can use the integral properties as follows:

Part (b)

Step 1: Apply the difference rule of integration

The difference rule states that \(\int_{a}^{b}(f(x)-g(x))dx=\int_{a}^{b}f(x)dx-\int_{a}^{b}g(x)dx\). For the integral \(\int_{3}^{?}(f(x) - g(x))dx\) (assuming the upper limit is missing, but if we assume the upper limit is \(5\) as in part (c) for consistency, let's assume the upper limit is \(5\) for part (b) as well, maybe a typo in the problem). So \(\int_{3}^{5}(f(x)-g(x))dx=\int_{3}^{5}f(x)dx-\int_{3}^{5}g(x)dx\)

Part (c)

Step 1: Apply the difference rule of integration

Using the difference rule \(\int_{3}^{5}(f(x)-g(x))dx=\int_{3}^{5}f(x)dx-\int_{3}^{5}g(x)dx\)

Part (d)

Step 1: Apply the difference rule and limit reversal

First, use the difference rule: \(\int_{5}^{7}(g(x)-f(x))dx=\int_{5}^{7}g(x)dx-\int_{5}^{7}f(x)dx\)

Part (e)

Step 1: Apply the constant multiple rule

Using the constant multiple rule \(\int_{5}^{7}7g(x)dx = 7\cdot\int_{5}^{7}g(x)dx\)

Part (f)

Step 1: Apply the constant multiple rule and limit reversal

First, reverse the limits of integration: \(\int_{5}^{3}4f(x)dx=-\int_{3}^{5}4f(x)dx\)
Then, apply the constant multiple rule: \(-\int_{3}^{5}4f(x)dx=- 4\cdot\int_{3}^{5}f(x)dx\)

Since the problem is incomplete (missing the necessary information about \(f(x)\) and \(g(x)\) or their integrals), we cannot provide numerical answers. Please provide the values of the integrals of \(f(x)\) and \(g(x)\) over the given intervals or the graphs of \(f(x)\) and \(g(x)\) to find the numerical values.

Answer:

To solve these integral problems, we need to recall the properties of definite integrals, such as the difference rule \(\int_{a}^{b}(f(x) - g(x))dx=\int_{a}^{b}f(x)dx-\int_{a}^{b}g(x)dx\), the constant multiple rule \(\int_{a}^{b}k\cdot f(x)dx = k\cdot\int_{a}^{b}f(x)dx\) (where \(k\) is a constant), and the property of reversing the limits of integration \(\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx\). However, since the problem does not provide the specific functions \(f(x)\) and \(g(x)\) or their integral values over the given intervals, we cannot compute the numerical answers. We need more information (such as the graphs of \(f(x)\) and \(g(x)\), the values of \(\int_{a}^{b}f(x)dx\) and \(\int_{a}^{b}g(x)dx\) over the intervals \([3,5]\), \([5,7]\), etc.) to proceed with the calculations.

If we assume that we have the values of \(\int_{3}^{5}f(x)dx = F_1\), \(\int_{3}^{5}g(x)dx=G_1\), \(\int_{5}^{7}f(x)dx = F_2\), \(\int_{5}^{7}g(x)dx=G_2\), \(\int_{3}^{7}f(x)dx=F_1 + F_2\) and \(\int_{3}^{7}g(x)dx=G_1+G_2\), we can use the integral properties as follows:

Part (b)

Step 1: Apply the difference rule of integration

The difference rule states that \(\int_{a}^{b}(f(x)-g(x))dx=\int_{a}^{b}f(x)dx-\int_{a}^{b}g(x)dx\). For the integral \(\int_{3}^{?}(f(x) - g(x))dx\) (assuming the upper limit is missing, but if we assume the upper limit is \(5\) as in part (c) for consistency, let's assume the upper limit is \(5\) for part (b) as well, maybe a typo in the problem). So \(\int_{3}^{5}(f(x)-g(x))dx=\int_{3}^{5}f(x)dx-\int_{3}^{5}g(x)dx\)

Part (c)

Step 1: Apply the difference rule of integration

Using the difference rule \(\int_{3}^{5}(f(x)-g(x))dx=\int_{3}^{5}f(x)dx-\int_{3}^{5}g(x)dx\)

Part (d)

Step 1: Apply the difference rule and limit reversal

First, use the difference rule: \(\int_{5}^{7}(g(x)-f(x))dx=\int_{5}^{7}g(x)dx-\int_{5}^{7}f(x)dx\)

Part (e)

Step 1: Apply the constant multiple rule

Using the constant multiple rule \(\int_{5}^{7}7g(x)dx = 7\cdot\int_{5}^{7}g(x)dx\)

Part (f)

Step 1: Apply the constant multiple rule and limit reversal

First, reverse the limits of integration: \(\int_{5}^{3}4f(x)dx=-\int_{3}^{5}4f(x)dx\)
Then, apply the constant multiple rule: \(-\int_{3}^{5}4f(x)dx=- 4\cdot\int_{3}^{5}f(x)dx\)

Since the problem is incomplete (missing the necessary information about \(f(x)\) and \(g(x)\) or their integrals), we cannot provide numerical answers. Please provide the values of the integrals of \(f(x)\) and \(g(x)\) over the given intervals or the graphs of \(f(x)\) and \(g(x)\) to find the numerical values.