Pls help i need it bad due to day
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The 4 subintervals are given: [2, 4], [4, 7], [7, 9], and [9, 10].
Each subinterval has length: 4 - 2 = 2, 7 - 4 = 3, 9 - 7 = 2, and 10 - 9 = 1.
Over each subinterval, we take the value of the function at the right endpoint: 3, 8, 15, and 18.
Then the integral is approximately
so 78.0 is the correct answer.
I love these. It's often called the Shoelace Formula. It actually works for the area of any 2D polygon.
We can derive it by first imagining our triangle in the first quadrant, one vertex at the origin, one at (a,b), one at (c,d), with (0,0),(a,b),(c,d) in counterclockwise order.
Our triangle is inscribed in the rectangle. There are three right triangles in that rectangle that aren't part of our triangle. When we subtract the area of the right triangles from the area of the rectangle we're left with the area S of our triangle.
That's the cross product in the purest form. When we're away from the origin, a arbitrary triangle with vertices will have the same area as one whose vertex C is translated to the origin.
We set
That's a perfectly useful formula right there. But it's usually multiplied out:
That's the usual form, the sum of cross products. Let's line up our numbers to make it easier.
(1, 2), (3, 4), (−7, 7)
(−7, 7),(1, 2), (3, 4),
[tex]A = \frac 1 2 ( 1(7)-2(-7) + 3(2)-4(1) + -7(4) - (7)(3)
