Answer:
area ≈ 9.219 square units
Explanation:
You want the approximate area under the curve y = -1/2x² +x +5 on the interval [1.5, 4] using 5 trapezoids.
Trapezoid area
The interval can be divided into 5 intervals of width ...
(4 -1.5)/5 = 2.5/5 = 0.5
The "bases" of each trapezoid will be the function values at the ends of the intervals, for example, at x=1.5 and x=2. The "height" of each trapezoid is the width of the sub-interval, 0.5.
The area formula for a trapezoid applies:
A = 1/2(b1 +b2)h
A = 1/2(f(x) +f(x +0.5))·0.5 . . . . . for x = 1.5, 2, 2.5, 3, 3.5
Approximate total area
The sum of the areas is computed in the attachment as ...
area under the curve = 9.21875
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Additional comment
The value of the integral is 445/48 ≈ 9.2708333...