Use calculus to find the area A of the triangle with the given vertices. (0, 0), (5, 3), (1, 4)

Answer 1

This Desmos graph shows some of the work involved in evaluating the integral setup in the video:

desmos.com/calculator/ikhblkjyza

Answer 2

I know that you have been asked to use calculus to solve this problem and 2 other tutors have shown you how this can be done.

Just for general information I want to point out that there is a determinant for solving this problem as well. It is difficult to write the determinant using this editor but the rows are as follow

x1 y1 1

x2 y2 1

x3 y3 1

Compute this determinant and multiply by 1/2.

The x's and y's are the co-ordinates of the 3 given points taken in the counterclockwise direction.

Just FYI!

Answer 3

Find the equations of the lines forming the sides of the triangle.

Line 1 (between (0,0) and (5,3)):

The slope is (3-0)/(5-0) = 3/5. Using the point-slope form, the equation is y - 0 = (3/5)(x - 0), or y = (3/5)x.

Line 2 (between (0,0) and (1,4)):

The slope is (4-0)/(1-0) = 4. The equation is y - 0 = 4(x - 0), or y = 4x.

Line 3 (between (5,3) and (1,4)):

The slope is (4-3)/(1-5) = 1/(-4) = -1/4. Using the point-slope form with (5,3): y - 3 = (-1/4)(x - 5), or y = (-1/4)x + 5/4 + 3 which simplifies to y = (-1/4)x + 17/4

2. Set up the integral.

The area of the triangle can be found by integrating the difference between the "upper" function and the "lower" function over the appropriate interval of x-values.

From x = 0 to x = 1, the upper function is y = 4x and the lower function is y = (3/5)x.

From x = 1 to x = 5, the upper function is y = (-1/4)x + 17/4 and the lower function is y = (3/5)x.

So, the area A is given by:

A = ∫₀¹ (4x - (3/5)x) dx + ∫₁⁵ ((-1/4)x + 17/4 - (3/5)x) dx

3. Evaluate the integrals.

First integral:

∫₀¹ (4x - (3/5)x) dx = ∫₀¹ (17/5)x dx = (17/5)(x²/2)|₀¹ = (17/10)(1² - 0²) = 17/10

Second integral:

∫₁⁵ ((-1/4)x + 17/4 - (3/5)x) dx = ∫₁⁵ ((-17/20)x + 17/4) dx = ((-17/20)(x²/2) + (17/4)x)|₁⁵ = (-17/40)(x²) + (17/4)x | from 1 to 5

= [(-17/40)(25) + (17/4)(5)] - [(-17/40)(1) + (17/4)(1)]

= [-85/8 + 85/4] - [-17/40 + 17/4]

= [-85/8 + 170/8] - [-17/40 + 170/40]

= 85/8 - 153/40

= 425/40 - 153/40 = 272/40 = 68/10 = 34/5

4. Add the results.

A = (17/10) + (34/5) = 17/10 + 68/10 = 85/10 = 8.5