Estimate the area under the curve f(x) = x2 from x = 1 to x = 5 by using four inscribed (under the curve) rectangles. Answer to the nearest integer.
The answers 30, dont ask how I got it
30 is the correct answer.
1 to 5 is 4 units, so each rectangle is one unit wide
the height of each rectangle is x^2
... x equals one to four ... under the curve
find the combined area
Shouldn't it specify of you are going from the left or the right?
To estimate the area under the curve using four inscribed rectangles, we'll use the Left Riemann Sum method. This involves dividing the interval [1, 5] into equal subintervals and approximating the area using rectangles that are inscribed under the curve.
Step 1: Calculate the width of each rectangle.
Since we are using four rectangles, we'll divide the interval [1, 5] into four equal subintervals. The width of each rectangle will be the length of each subinterval.
Interval width = (upper limit - lower limit)/(number of rectangles)
Interval width = (5 - 1)/4 = 1
Step 2: Calculate the height of each rectangle.
To find the height of each rectangle, we substitute the x-values at the left endpoints of each subinterval into the function f(x) = x^2.
For our four subintervals, the left endpoints are 1, 2, 3, and 4.
Height of first rectangle = f(1) = 1^2 = 1
Height of second rectangle = f(2) = 2^2 = 4
Height of third rectangle = f(3) = 3^2 = 9
Height of fourth rectangle = f(4) = 4^2 = 16
Step 3: Calculate the area of each rectangle.
To find the area of each rectangle, multiply the width by the height.
Area of first rectangle = 1 * 1 = 1
Area of second rectangle = 1 * 4 = 4
Area of third rectangle = 1 * 9 = 9
Area of fourth rectangle = 1 * 16 = 16
Step 4: Sum up the areas of all four rectangles.
Total approximate area = Area of first rectangle + Area of second rectangle + Area of third rectangle + Area of fourth rectangle
Total approximate area = 1 + 4 + 9 + 16 = 30
Therefore, the estimated area under the curve f(x) = x^2 from x = 1 to x = 5, using four inscribed rectangles, is approximately 30 square units.