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Math  /  Calculus

Question7. [-/1 Points] DETAILS MY NOTES SESSCALCET2 7.2.029.
A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced 1.5 cm apart. The liver is 15 cm long and the cross-sectional areas, in square centimeters, are 0,17,58,79,93,107,116,128,63,380,17,58,79,93,107,116,128,63,38, and 0 . Use the Midpoint Rule with n=5n=5 to estimate the volume VV of the liver. v=v= \qquad cm3\mathrm{cm}^{3} Need Help? Watch 17 Submil Answer

Studdy Solution

STEP 1

1. The liver is modeled by cross-sectional areas measured at regular intervals.
2. The cross-sectional areas are given at intervals of 1.5 cm.
3. The total length of the liver is 15 cm.
4. We are using the Midpoint Rule with n=5 n = 5 to estimate the volume.

STEP 2

1. Determine the width of each subinterval.
2. Identify the midpoints of each subinterval.
3. Calculate the volume using the Midpoint Rule.

STEP 3

The total length of the liver is 15 cm, and we need to divide this into 5 subintervals. Therefore, the width of each subinterval is:
15 cm5=3 cm \frac{15 \, \text{cm}}{5} = 3 \, \text{cm}

STEP 4

The cross-sectional areas are given at intervals of 1.5 cm. We need to find the midpoints of each 3 cm subinterval:
- First subinterval: Midpoint at 1.5 cm - Second subinterval: Midpoint at 4.5 cm - Third subinterval: Midpoint at 7.5 cm - Fourth subinterval: Midpoint at 10.5 cm - Fifth subinterval: Midpoint at 13.5 cm

STEP 5

Using the Midpoint Rule, the volume V V is estimated by:
V≈∑i=15A(xi∗)⋅Δx V \approx \sum_{i=1}^{5} A(x_i^*) \cdot \Delta x
where A(xi∗) A(x_i^*) is the area at the midpoint of each subinterval and Δx=3 cm \Delta x = 3 \, \text{cm} .
The areas at the midpoints are: 17, 79, 107, 128, and 38 square centimeters.
V≈(17+79+107+128+38)×3 V \approx (17 + 79 + 107 + 128 + 38) \times 3
V≈369×3 V \approx 369 \times 3
V≈1107 cm3 V \approx 1107 \, \text{cm}^3
The estimated volume of the liver is:
1107 cm3 \boxed{1107 \, \text{cm}^3}

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