Question: The graph of \( f \) is shown. Evaluate each integral. (a) \(\int_{0}^{18} f(x) \, dx\) (b)…

The graph of \( f \) is shown. Evaluate each integral.

(a) \(\int_{0}^{18} f(x) \, dx\)

(b) \(\int_{0}^{45} f(x) \, dx\)

(c) \(\int_{45}^{63} f(x) \, dx\)

(d) \(\int_{0}^{81} f(x) \, dx\)

The graph shows the function \( y = f(x) \) with the following points:

• \( (0, 0) \)
• \( (18, 18) \)
• \( (36, 18) \)
• \( (54, 0) \)
• \( (72, 0) \)

The area under the curve is divided into sections, and the values are:

• From \( x = 0 \) to \( x = 18 \), the area is 324.

Solution

Problem (b): \(\int_{0}^{45} f(x) \, dx\) Step 1: Break the integral into intervals where \( f(x) \) is defined differently. \[ \int_{0}^{45} f(x) \, dx = \int_{0}^{18} 2x \, dx + \int_{18}^{36} 36 \, dx + \int_{36}^{45} (-2x + 108) \, dx \] Step 2: Evaluate each integral separately. \[ \int_{0}^{18} 2x \, dx = \left[ x^2 \right]{0}^{18} = 18^2 - 0^2 = 324 \] \[ \int{18}^{36} 36 \, dx = 36 \times (36 - 18) = 36 \times 18 = 648 \] \[ \int_{36}^{45} (-2x + 108) \, dx = \left[ -x^2 + 108x \right]{36}^{45} = (-45^2 + 108 \times 45) - (-36^2 + 108 \times 36) = ( -2025 + 4860 ) - ( -1296 + 3888 ) = 2835 - 2592 = 243 \] Step 3: Sum all the results. \[ 324 + 648 + 243 = 1215 \] Answer: \(1215\) --- Problem (c): \(\int{45}^{68} f(x) \, dx\) Step 1: Determine the intervals where \( f(x) \) is defined. \[ \int_{45}^{68} f(x) \, dx = \int_{45}^{54} (-2x + 108) \, dx + \int_{54}^{68} 0 \, dx \] Step 2: Evaluate each integral separately. \[ \int_{45}^{54} (-2x + 108) \, dx = \left[ -x^2 + 108x \right]{45}^{54} = (-54^2 + 108 \times 54) - (-45^2 + 108 \times 45) = ( -2916 + 5832 ) - ( -2025 + 4860 ) = 2916 - 2835 = 81 \] \[ \int{54}^{68} 0 \, dx = 0 \] Step 3: Sum all the results. \[ 81 + 0 = 81 \] Answer: \(81\) --- Problem (d): \(\int^{81} f(x) \, dx\) (Assuming the lower limit is 0 based on the context) Step 1: Break the integral into intervals where \( f(x) \) is defined differently. \[ \int_{0}^{81} f(x) \, dx = \int_{0}^{18} 2x \, dx + \int_{18}^{36} 36 \, dx + \int_{36}^{54} (-2x + 108) \, dx + \int_{54}^{81} 0 \, dx \] Step 2: Evaluate each integral separately. \[ \int_{0}^{18} 2x \, dx = \left[ x^2 \right]{0}^{18} = 18^2 - 0^2 = 324 \] \[ \int{18}^{36} 36 \, dx = 36 \times (36 - 18) = 36 \times 18 = 648 \] \[ \int_{36}^{54} (-2x + 108) \, dx = \left[ -x^2 + 108x \right]{36}^{54} = (-54^2 + 108 \times 54) - (-36^2 + 108 \times 36) = ( -2916 + 5832 ) - ( -1296 + 3888 ) = 2916 - 2592 = 324 \] \[ \int{54}^{81} 0 \, dx = 0 \] Step 3: Sum all the results. \[ 324 + 648 + 324 + 0 = 1296 \] Answer: \(1296\)