using 5 rectangles what is the area under a curve using the function f(x)=3x+4 with boundries at [0,2]
Assuming equal subintervals, the width of each is 0.4
Using left endpoints, the heights are
4,5.2,6.4,...
Now just add up the areas of the rectangles
A nice online calculator for this can be found at
http://mathworld.wolfram.com/RiemannSum.html
To find the area under the curve represented by the function f(x) = 3x + 4 using 5 rectangles, we can apply the method of Riemann sums. The area estimation can be calculated by dividing the interval [0, 2] into 5 equal subintervals and approximating the height of each rectangle using the function values at specific points within those subintervals.
Step 1: Divide the interval [0, 2] into 5 equal subintervals:
Each subinterval will have a width of (2 - 0) / 5 = 0.4
Step 2: Determine the x-values at which to evaluate the function:
To obtain the height of each rectangle, we need to choose representative x-values within each subinterval. A widely-used approach is to use the left endpoints of the subintervals.
For 5 subintervals with a width of 0.4, the x-values to evaluate the function will be:
x1 = 0
x2 = 0.4
x3 = 0.8
x4 = 1.2
x5 = 1.6
Step 3: Evaluate the function at the chosen x-values:
Evaluate f(x) = 3x + 4 at each of the selected x-values:
f(x1) = 3(0) + 4 = 4
f(x2) = 3(0.4) + 4 = 4.2
f(x3) = 3(0.8) + 4 = 6.4
f(x4) = 3(1.2) + 4 = 7.6
f(x5) = 3(1.6) + 4 = 8.8
Step 4: Calculate the area of each rectangle:
The height of each rectangle will be the function value at the corresponding x-value. The width of each rectangle is the width of the subintervals, which is 0.4.
Area of rectangle 1 = height * width = f(x1) * 0.4 = 4 * 0.4 = 1.6
Area of rectangle 2 = f(x2) * 0.4 = 4.2 * 0.4 = 1.68
Area of rectangle 3 = f(x3) * 0.4 = 6.4 * 0.4 = 2.56
Area of rectangle 4 = f(x4) * 0.4 = 7.6 * 0.4 = 3.04
Area of rectangle 5 = f(x5) * 0.4 = 8.8 * 0.4 = 3.52
Step 5: Sum up the areas of all the rectangles:
Total area ≈ Area of rectangle 1 + Area of rectangle 2 + Area of rectangle 3 + Area of rectangle 4 + Area of rectangle 5
≈ 1.6 + 1.68 + 2.56 + 3.04 + 3.52
≈ 12.4
Therefore, using 5 rectangles, the approximate area under the curve represented by the function f(x) = 3x + 4 between the boundaries of 0 and 2 is approximately 12.4 square units.