1. ## Intergral Problems!!!

1. Evaluate the integral by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry.

2. Evaluate the integral below by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry.

3. Evaluate the definite integral by interpreting it in terms of areas.

2. Originally Posted by Kayla_N
1. Evaluate the integral by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry.

Can you draw this region? It is just a line with a slope of 10 and $y$-intercept at $(0,\,-3),$ except that the negative portion is flipped up above the $x$-axis because of the absolute value. The region is just two triangles.

2. Evaluate the integral below by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry.

What is the graph of $x^2+y^2=r^2?$ And if you solve for $y,$ ignoring the negative root, what then?

Spoiler:
It is the upper half of a circle. You should be familiar with the standard equation of a circle.
3. Evaluate the definite integral by interpreting it in terms of areas.

Draw the region. It is two triangles. Remember that area below the $x$-axis is subtracted.

3. 3. Evaluate the definite integral by interpreting it in terms of areas.

i did graph and only 1 triangle show up. x= 4 and y= -21

4. Originally Posted by Kayla_N
3. Evaluate the definite integral by interpreting it in terms of areas.

i did graph and only 1 triangle show up. x= 4 and y= -21
Are you sure you drew the graph correctly? It is just a line with $x$-intercept at $(4,\,0).$ On $[3,\,6],$ part of the line will be above the $x$-axis, and part will be below. This gives you two right triangles (the line forms the hypotenuses, the $x$-axis forms one leg of each, and the lines $x=3$ and $x=6$ form the other legs).

5. Well i might drew it wrong. But question.. dont you have to take the antiderivative of (5x-20) first?? I read your post, so the area of triangle is b*h= 3*6 =18?

6. Originally Posted by Kayla_N
Well i might drew it wrong. But question.. dont you have to take the antiderivative of (5x-20) first??
The definite integral of a curve is the area below the curve (except where the curve is negative, in which case any area is subtracted). So to find the definite integral of $5x-20,$ graph it and calculate the area using basic geometric formulae.

You only take the antiderivative if you want to use it to apply the fundamental theorem of calculus, but the problem tells you to use geometric formulae instead.

I read your post, so the area of triangle is b*h= 3*6 =18?
Where did you get those values? And the formula for the area of a triangle is $A=\frac12bh.$

7. Originally Posted by Kayla_N
Well i might drew it wrong. But question.. dont you have to take the antiderivative of (5x-20) first?
Since the integral is the area, integrating and then graphing and finding the area (or trying to) would not only be difficult, but would give you the wrong answer.

Instead, try following the instructions. First, review your book and your class notes for their discussions of the "area" concept as it relates to integration.

Then review how to graph linear functions, radical functions, and absolute-value functions.

Once you've learned that material, draw the pictures for your three integrands. One should be a right triangle, one should be a half-circle, and one should be an isosceles triangle.

Use the formulas for the area of a triangle with base length "b" and height "h" and the area of a circle with radius "r" to find the areas "using geometrical methods".

8. Thanks Everyone for helping out..I solved all the problems already. Thank you very much .