use trapezoidal rule to estimate the area of the equation x^2+y^2=9. Use trapeziums of width 1
To estimate the area of the equation x^2 + y^2 = 9 using the trapezoidal rule, we first need to rewrite the equation in terms of y to obtain y as a function of x.
x^2 + y^2 = 9
y^2 = 9 - x^2
y = ± sqrt(9 - x^2)
Since we are interested in the area where y is positive, we will use y = sqrt(9 - x^2).
Next, we will set up trapeziums of width 1 and compute the areas of each trapezium to estimate the total area under the curve.
Let's set up the trapeziums of width 1 in the x-direction, where x ranges from -3 to 3.
The first trapezium has a base of 1 and heights of sqrt(9 - (-3)^2) = sqrt(9-9) = 0 and sqrt(9 - (-2)^2) = sqrt(9-4) = sqrt(5).
The area of the first trapezium is (0 + sqrt(5))/2 * 1 = sqrt(5)/2.
Continuing this process for the rest of the trapeziums, we can estimate the total area under the curve by summing the areas of all the trapeziums.
The total area ≈ sqrt(5)/2 + sqrt(8)/2 + sqrt(5)/2 + ... = 4(sqrt(5) + sqrt(8) + sqrt(5))/2 = 9(sqrt(5) + sqrt(8))/2 ≈ 11.22 units^2
Therefore, using the trapezoidal rule, the estimated area of the equation x^2 + y^2 = 9 is approximately 11.22 units^2.