again, each rectangle has width 1/2, so the area is
1/2 (f(2) + f(5/2) + ... + f(9/2))
The area is approximately=
just answer and explanation please and thanks.
1/2 (f(2) + f(5/2) + ... + f(9/2))
Δx = (5 - 2) / 6
= 3 / 6
= 0.5
Next, we calculate the left endpoint of each subinterval, starting from x = 2 and incrementing by Δx:
x1 = 2
x2 = 2 + 0.5
= 2.5
x3 = 2.5 + 0.5
= 3
x4 = 3 + 0.5
= 3.5
x5 = 3.5 + 0.5
= 4
x6 = 4 + 0.5
= 4.5
Then, we evaluate the function at each left endpoint to find the corresponding y-values:
y1 = (x1)^3 + 2
= (2)^3 + 2
= 8 + 2
= 10
y2 = (x2)^3 + 2
= (2.5)^3 + 2
= 15.625 + 2
= 17.625
y3 = (x3)^3 + 2
= (3)^3 + 2
= 27 + 2
= 29
y4 = (x4)^3 + 2
= (3.5)^3 + 2
= 42.875 + 2
= 44.875
y5 = (x5)^3 + 2
= (4)^3 + 2
= 64 + 2
= 66
y6 = (x6)^3 + 2
= (4.5)^3 + 2
= 91.125 + 2
= 93.125
Finally, we calculate the sum of the areas of the rectangles formed by each subinterval by multiplying the width of each subinterval by its corresponding height and adding them together:
Area ≈ (Δx) * (y1 + y2 + y3 + y4 + y5 + y6)
= 0.5 * (10 + 17.625 + 29 + 44.875 + 66 + 93.125)
= 0.5 * 260.625
= 130.3125
Therefore, the area under the curve y = x^3 + 2 from x = 2 to x = 5, approximated using a left sum with 6 subdivisions, is approximately 130.3125 square units.
Δx = (5 - 2) / 6 = 3 / 6 = 0.5
Next, we need to evaluate the function at the left endpoint of each subdivision and multiply it by the width (Δx). The left endpoint of the first subdivision is x = 2. For subsequent subdivisions, we add Δx to the previous left endpoint. So, the left endpoints for the 6 subdivisions are:
x1 = 2
x2 = 2 + Δx
x3 = 2 + 2Δx
x4 = 2 + 3Δx
x5 = 2 + 4Δx
x6 = 2 + 5Δx
Since the function is y = x^3 + 2, we can substitute the values of x into the function to find the corresponding y-values.
Once we have the y-values, we calculate the area by summing up the products of each y-value and the width Δx:
Area ≈ (y1 * Δx) + (y2 * Δx) + (y3 * Δx) + (y4 * Δx) + (y5 * Δx) + (y6 * Δx)
Finally, we compute the approximation by plugging in the values:
Area ≈ [(2^3 + 2) * 0.5] + [(2.5^3 + 2) * 0.5] + [(3^3 + 2) * 0.5] + [(3.5^3 + 2) * 0.5] + [(4^3 + 2) * 0.5] + [(4.5^3 + 2) * 0.5]