Find the area of the region bounded by the graph of y^3=x^3 and the chord joining the points (-1, 1) and (8,
To find the area of the region bounded by the graph of y^3 = x^3 and the chord joining the points (-1, 1) and (8, ?), we need to integrate the function and subtract the areas below the x-axis.
Let's start by finding the intersection points of the graph and the chord.
First, we need to find the y-coordinate of the point (8, ?) on the chord. Using the equation for the line passing through two points, we can find the slope (m) of the line:
m = (y2 - y1) / (x2 - x1) = ( ? - 1) / (8 - (-1)) = ( ? - 1) / 9
Since the line passes through (8, ?), we can substitute the x and y values into the equation of the line to find the y-coordinate:
? = m(x - x1) + y1 = ( ? - 1) / 9 (x - 8) + 1
Simplifying the equation, we have:
? / 9 * (x - 8) = 1 - 1/9
? / 9 * (x - 8) = 8/9
Next, we need to find the intersection points of the graph y^3 = x^3 and the chord. We can equate the y-values of the graph and the chord:
y^3 = ? / 9 * (x - 8)
Taking the cube root of both sides, we have:
y = ? / 3 * ∛(x - 8)
Now, let's find the intersection points.
Substituting this expression for y into the equation of the graph, we have:
( ? / 3 * ∛(x - 8))^3 = x^3
Simplifying, we get:
( ?^3 / 27) * (x - 8) = x^3
?^3 * (x - 8) = 27x^3
?^3 * x - 8?^3 = 27x^3
(1 - 27)x^3 = 8?^3
26x^3 = 8?^3
x = (8?^3) / 26
x = (4?^3) / 13
Now that we have the x-coordinate of the intersection point, we can substitute it back into the equation of the chord to find the y-coordinate:
? = ( ? - 1) / 9 ((4?^3) / 13 - 8) + 1
Simplifying, we get:
? / 9 * ((4?^3) / 13 - 8) = 8 / 9 - 1
? / 9 * ((4?^3) / 13 - 8) = -1 / 9
Solving for ?, we get:
? = -1 * 9 * (4?^3 / 13 - 8)
? = -9 * (4?^3 / 13 - 8)
? = -9 * ((4?^3 - 104) / 13)
13? = -36?^3 + 936
36?^3 + 13? - 936 = 0
To find the numeric value of ?, we can solve this cubic equation using numerical methods or approximations.
Once we have the numeric value of ?, we can calculate the points of intersection and proceed to find the area.