Please complete answering question 29 and SHOW ALL WORK! Thanks. y = 1/4x2, y =

Question

Please complete answering question 29 and SHOW ALL WORK! Thanks.

y = 1/4x2, y = 2x2, x + y = 3, x 0 Use calculus to find the area of the triangle with the given vertices. (0,0), (3,1), (1.2) 30. (2,0), (0.2), ( - 1,1) Evaluate the integral and interpret it as the area of a region. Sketch the region. pi/2 0|sin - cos 2x | dx 4 0 | x + 2 - x | dx Use a graph to find approximate x - coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.

Explanation / Answer

Step 1: Graph the triangle.

Draw a coordinate system and plot the given points on the graph. Label the vertices as follows: O=(0,0), A=(3,1), and B=(1,2). Connect the dots to form the triangle: OA, OB, and AB.

By "use Calculus" to find the area, I'm going to assume that the following technique is used: Determine the area under the graph of AB, and then subtract from that the areas under the graphs of OA and OB. What you're left with is the area of the triangle.


Step 2: Find equations of the lines containing the three sides of the triangle.

I'm going to assume you know how to find the slope and y-intercept of lines, and will omit it here; but when you're done, you'll have:

AB: y = (-1/2)x + 5/2
OA: y = (1/3)x
OB: y = 2x


Step 3: Integrate.

First determine the intervals of integration. For AB, you're integrating from 1 to 3. For OA, integrate from 0 to 3. Finally, for OB integrate from 0 to 1.

Area of triangle =
INTEGRAL[1,3] [ (-1/2)x + 3/2 ]dx - INTEGRAL[0,3] (1/3)x dx - INTEGRAL[0,1] (2x)dx =

2.5

So the triangle has an area of 2.5, by my calculations.

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