Three rectangles are used to estimate the area under the curve, y=x2+4 , from x = 0 to x = 6. Using the right-hand rectangular approximation method (RRAM), what is the approximate area under the curve for the given interval? (1 point) Responses 47 47 64 64 68 68 94 94 136
To estimate the area using the right-hand rectangular approximation method (RRAM), we divide the interval [0, 6] into three equal-sized subintervals of width 2.
The height of each rectangle is determined by the right endpoint of each subinterval.
For the first subinterval [0, 2], the right endpoint is x = 2.
Using this value in the equation y = x^2 + 4, we get y = 2^2 + 4 = 8.
So, the height of the first rectangle is 8.
For the second subinterval [2, 4], the right endpoint is x = 4.
Using this value in the equation y = x^2 + 4, we get y = 4^2 + 4 = 20.
So, the height of the second rectangle is 20.
For the third subinterval [4, 6], the right endpoint is x = 6.
Using this value in the equation y = x^2 + 4, we get y = 6^2 + 4 = 40.
So, the height of the third rectangle is 40.
The width of each rectangle is 2 (the width of each subinterval).
Therefore, the area of each rectangle is the product of its width and height:
Area of first rectangle = 2 * 8 = 16
Area of second rectangle = 2 * 20 = 40
Area of third rectangle = 2 * 40 = 80
The approximate area under the curve using RRAM is the sum of the areas of the three rectangles:
16 + 40 + 80 = 136
So, the approximate area under the curve for the given interval is 136.