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To approximate the area under the curve of the function [tex]\( f(x) = 3x^2 + 1 \)[/tex] from [tex]\( x = 0 \)[/tex] to [tex]\( x = 4 \)[/tex] using the trapezoidal rule with 4 subintervals, we will follow these steps:
1. Divide the interval [tex]\([0, 4]\)[/tex] into 4 subintervals:
The interval [tex]\([0, 4]\)[/tex] has a total length of [tex]\( 4 \)[/tex]. Dividing this into 4 equal subintervals, each subinterval will have a width of:
[tex]\[ \Delta x = \frac{4 - 0}{4} = 1 \][/tex]
2. Calculate the x-values at the boundaries of each subinterval:
The x-values will be:
[tex]\[ x_0 = 0, \quad x_1 = 1, \quad x_2 = 2, \quad x_3 = 3, \quad x_4 = 4 \][/tex]
3. Evaluate the function [tex]\( f(x) \)[/tex] at these x-values:
[tex]\[ f(0) = 3(0)^2 + 1 = 1 \][/tex]
[tex]\[ f(1) = 3(1)^2 + 1 = 4 \][/tex]
[tex]\[ f(2) = 3(2)^2 + 1 = 13 \][/tex]
[tex]\[ f(3) = 3(3)^2 + 1 = 28 \][/tex]
[tex]\[ f(4) = 3(4)^2 + 1 = 49 \][/tex]
4. Calculate the areas of the trapezoids for each subinterval:
Using the trapezoidal rule, the area of a trapezoid for each subinterval can be calculated by:
[tex]\[ \text{Area} = \frac{\Delta x}{2} \times (f(x_i) + f(x_{i+1})) \][/tex]
- For the interval [tex]\([0, 1]\)[/tex]:
[tex]\[ \text{Area}_1 = \frac{1}{2} \times (1 + 4) = 2.5 \][/tex]
- For the interval [tex]\([1, 2]\)[/tex]:
[tex]\[ \text{Area}_2 = \frac{1}{2} \times (4 + 13) = 8.5 \][/tex]
- For the interval [tex]\([2, 3]\)[/tex]:
[tex]\[ \text{Area}_3 = \frac{1}{2} \times (13 + 28) = 20.5 \][/tex]
- For the interval [tex]\([3, 4]\)[/tex]:
[tex]\[ \text{Area}_4 = \frac{1}{2} \times (28 + 49) = 38.5 \][/tex]
5. Sum up the areas of each trapezoid to get the total estimated area:
[tex]\[ \text{Total Area} = 2.5 + 8.5 + 20.5 + 38.5 = 70.0 \][/tex]
Thus, the approximate area of the region enclosed by the graph of [tex]\( f(x) = 3x^2 + 1 \)[/tex], the [tex]\( x \)[/tex]-axis, [tex]\( x = 0 \)[/tex], and [tex]\( x = 4 \)[/tex] is [tex]\(\boxed{70.0}\)[/tex].
Reviewing the options given in the question:
A. 46
B. 49
C. 94
D. 184
None of these options matches the computed result of [tex]\( 70.0 \)[/tex]. Therefore, an error seems to have been made in formulating the choices, as the correct approximate area is [tex]\( \boxed{70.0} \)[/tex].
1. Divide the interval [tex]\([0, 4]\)[/tex] into 4 subintervals:
The interval [tex]\([0, 4]\)[/tex] has a total length of [tex]\( 4 \)[/tex]. Dividing this into 4 equal subintervals, each subinterval will have a width of:
[tex]\[ \Delta x = \frac{4 - 0}{4} = 1 \][/tex]
2. Calculate the x-values at the boundaries of each subinterval:
The x-values will be:
[tex]\[ x_0 = 0, \quad x_1 = 1, \quad x_2 = 2, \quad x_3 = 3, \quad x_4 = 4 \][/tex]
3. Evaluate the function [tex]\( f(x) \)[/tex] at these x-values:
[tex]\[ f(0) = 3(0)^2 + 1 = 1 \][/tex]
[tex]\[ f(1) = 3(1)^2 + 1 = 4 \][/tex]
[tex]\[ f(2) = 3(2)^2 + 1 = 13 \][/tex]
[tex]\[ f(3) = 3(3)^2 + 1 = 28 \][/tex]
[tex]\[ f(4) = 3(4)^2 + 1 = 49 \][/tex]
4. Calculate the areas of the trapezoids for each subinterval:
Using the trapezoidal rule, the area of a trapezoid for each subinterval can be calculated by:
[tex]\[ \text{Area} = \frac{\Delta x}{2} \times (f(x_i) + f(x_{i+1})) \][/tex]
- For the interval [tex]\([0, 1]\)[/tex]:
[tex]\[ \text{Area}_1 = \frac{1}{2} \times (1 + 4) = 2.5 \][/tex]
- For the interval [tex]\([1, 2]\)[/tex]:
[tex]\[ \text{Area}_2 = \frac{1}{2} \times (4 + 13) = 8.5 \][/tex]
- For the interval [tex]\([2, 3]\)[/tex]:
[tex]\[ \text{Area}_3 = \frac{1}{2} \times (13 + 28) = 20.5 \][/tex]
- For the interval [tex]\([3, 4]\)[/tex]:
[tex]\[ \text{Area}_4 = \frac{1}{2} \times (28 + 49) = 38.5 \][/tex]
5. Sum up the areas of each trapezoid to get the total estimated area:
[tex]\[ \text{Total Area} = 2.5 + 8.5 + 20.5 + 38.5 = 70.0 \][/tex]
Thus, the approximate area of the region enclosed by the graph of [tex]\( f(x) = 3x^2 + 1 \)[/tex], the [tex]\( x \)[/tex]-axis, [tex]\( x = 0 \)[/tex], and [tex]\( x = 4 \)[/tex] is [tex]\(\boxed{70.0}\)[/tex].
Reviewing the options given in the question:
A. 46
B. 49
C. 94
D. 184
None of these options matches the computed result of [tex]\( 70.0 \)[/tex]. Therefore, an error seems to have been made in formulating the choices, as the correct approximate area is [tex]\( \boxed{70.0} \)[/tex].
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