use trapezium rule with 3 intervals to estimate the value of intergrating x=0 and x=3 squarerooting 1+xsquared
Did you make your diagram?
I read your question as estimating the definite integral of √(1+x^2) from 0 to 3
You should be looking at 3 trapezoids with a base width of 1 each and the heights for the values of x = 0,1,2,3 would be 1,√2,√5, and √10 respectively.
The estimated area would be
(1+√2)(1)/2 + (√2+√5)(1)/2 + (√5+√10)(1)/2
= (1/2)(1 + 2√2 + 2√5 + √10)
= 5.73
check my arithmetic.
To estimate the value of the integral ∫(0 to 3) √(1+x^2) using the trapezium rule with 3 intervals, follow these steps:
Step 1: Divide the interval [0,3] into 3 equal sub-intervals.
The width, h, of each sub-interval can be calculated as:
h = (b - a) / n,
where b is the upper limit (3), a is the lower limit (0), and n is the number of intervals (3).
In this case, h = (3 - 0) / 3 = 1.
Step 2: Calculate the function values at each endpoint and midpoint of the sub-intervals.
Evaluate the function, √(1+x^2), at the endpoints and midpoints of the sub-intervals to get the corresponding function values.
For the 3 intervals, we have:
x0 = 0, x1 = 1, x2 = 2, x3 = 3.
Correspondingly, the function values are:
f(x0) = √(1+0^2) = 1,
f(x1) = √(1+1^2) = √2,
f(x2) = √(1+2^2) = √5,
f(x3) = √(1+3^2) = √10.
Step 3: Apply the trapezium rule formula.
The trapezium rule formula for approximating the integral is:
∫(a to b) f(x) dx ≈ h/2 * [f(x0) + 2 * (f(x1) + f(x2) + f(x3)) + f(x4)],
where h is the interval width and f(xi) are the function values calculated in Step 2.
Substituting the values, we get:
∫(0 to 3) √(1+x^2)dx ≈ 1/2 * [1 + 2 * (√2 + √5 + √10) + √10].
Step 4: Calculate the estimated value.
Evaluate the expression to obtain the estimated value:
∫(0 to 3) √(1+x^2)dx ≈ 1/2 * [1 + 2 * (√2 + √5 + √10) + √10]
≈ 1/2 * [1 + 2 * (1.414 + 2.236 + 3.162) + 3.162]
≈ 1/2 * [1 + 2 * 6.812 + 3.162]
≈ 1/2 * [1 + 13.624 + 3.162]
≈ 1/2 * 17.786
≈ 8.893.
To estimate the value of the definite integral using the trapezium rule, you'll need to follow these steps:
1. Determine the number of intervals:
In this case, you are asked to use 3 intervals. The interval width (h) can be calculated by dividing the total interval length by the number of intervals. Since the interval runs from x=0 to x=3, the interval width (h) is given by (3 - 0) / 3 = 1.
2. Calculate the function values:
Evaluate the function, √(1 + x^2), at the boundaries and the midpoints of each interval. In this case, calculate the function values at x = 0, 1, 2, and 3.
3. Apply the trapezium rule:
The trapezium rule formula is:
∫[a, b] f(x) dx ≈ h/2 * [f(a) + 2 * f(x₁) + 2 * f(x₂) + ... + f(b)]
where a and b are the lower and upper limits of integration, respectively.
Using the values calculated above, we can apply the formula as follows:
∫[0, 3] √(1 + x^2) dx ≈ (1/2) * [f(0) + 2 * f(1) + 2 * f(2) + f(3)]
Substituting the function values, we get:
≈ (1/2) * [f(0) + 2 * f(1) + 2 * f(2) + f(3)]
≈ (1/2) * [√(1 + 0^2) + 2 * √(1 + 1^2) + 2 * √(1 + 2^2) + √(1 + 3^2)]
Now, substitute the values into the equation and calculate the result:
≈ (1/2) * [√1 + 2 * √2 + 2 * √5 + √10]
≈ (1/2) * [1 + 2 * 1.414 + 2 * 2.236 + 3.162]
≈ (1/2) * [1 + 2.828 + 4.472 + 3.162]
≈ 5.737
Therefore, using the trapezium rule with 3 intervals, the estimate for the definite integral of √(1 + x^2) from x=0 to x=3 is approximately 5.737.