The estimated value of the integral from 0 to 2 of x^2 dx , using the trapezoidal rule with 4 trapezoids is
a) 2.75
b) 5.50
c) 1.88
d) 3.75
2.75
I took the test its correct
To estimate the value of the given integral using the trapezoidal rule with 4 trapezoids, we can follow these steps:
Step 1: Determine the width or size of each trapezoid by dividing the interval length (2 - 0 = 2) by the number of trapezoids (4).
Width = (b - a) / n
Width = (2 - 0) / 4
Width = 0.5
Step 2: Calculate the heights or function values at the endpoints and midpoints of each trapezoid. In this case, we need to evaluate the function x^2 at values 0, 0.5, 1, 1.5, and 2.
f(0) = 0^2 = 0
f(0.5) = (0.5)^2 = 0.25
f(1) = 1^2 = 1
f(1.5) = (1.5)^2 = 2.25
f(2) = 2^2 = 4
Step 3: Use the trapezoidal rule formula to calculate the approximate integral value.
Approximate Integral = (width / 2) * [f(a) + 2 * Σ f(xi) + f(b)]
Approximate Integral = (0.5 / 2) * [f(0) + 2 * (f(0.5) + f(1) + f(1.5)) + f(2)]
Approximate Integral = (0.25) * [0 + 2 * (0.25 + 1 + 2.25) + 4]
Approximate Integral = (0.25) * [0 + 2 * 3.5 + 4]
Approximate Integral = (0.25) * [0 + 7 + 4]
Approximate Integral = (0.25) * 11
Approximate Integral = 2.75
Therefore, the estimated value of the integral from 0 to 2 of x^2 dx, using the trapezoidal rule with 4 trapezoids, is 2.75.
Hence, the correct answer is a) 2.75.
To estimate the value of the integral using the trapezoidal rule, we divide the interval from 0 to 2 into equal subintervals and approximate the area under the curve using trapezoids.
The first step is to determine the width of each subinterval. In this case, we have 4 trapezoids, so there would be 5 equally spaced points: 0, 0.5, 1, 1.5, and 2. The width of each subinterval is then (2 - 0) / 4 = 0.5.
Next, we evaluate the function f(x) = x^2 at these 5 points. Plugging in the values, we get:
f(0) = 0^2 = 0,
f(0.5) = (0.5)^2 = 0.25,
f(1) = 1^2 = 1,
f(1.5) = (1.5)^2 = 2.25,
f(2) = 2^2 = 4.
Now, we calculate the area of each trapezoid using the formula: [(b1 + b2) / 2] * h, where b1 and b2 are the bases, and h is the height (or width) of the trapezoid.
For the first trapezoid:
b1 = f(0) = 0,
b2 = f(0.5) = 0.25,
h = 0.5.
Area1 = [(0 + 0.25) / 2] * 0.5 = 0.125.
For the second trapezoid:
b1 = f(0.5) = 0.25,
b2 = f(1) = 1,
h = 0.5.
Area2 = [(0.25 + 1) / 2] * 0.5 = 0.375.
For the third trapezoid:
b1 = f(1) = 1,
b2 = f(1.5) = 2.25,
h = 0.5.
Area3 = [(1 + 2.25) / 2] * 0.5 = 1.125.
For the fourth trapezoid:
b1 = f(1.5) = 2.25,
b2 = f(2) = 4,
h = 0.5.
Area4 = [(2.25 + 4) / 2] * 0.5 = 1.875.
Finally, we add up the areas of all the trapezoids to get the estimated value of the integral:
Estimated value = Area1 + Area2 + Area3 + Area4 = 0.125 + 0.375 + 1.125 + 1.875 = 3.5.
Since none of the given answer choices is exactly 3.5, we can round the estimated value to the nearest decimal place.
The closest answer choice to 3.5 is 3.75 (option d).
area = appr .5(0+.25) + .5(.25 + 1) + .5(1 + 2.25) + .5(2.25+4)
= .5(0 + .25 + .25 + 1 + 1 + 2.25 + 2.25 + 4)
= 5.5