Find the area of the region bounded by the parabola y = 5x^2, the tangent line to this parabola at (3, 45), and the x-axis.
To find the area of the region bounded by the parabola, the tangent line, and the x-axis, we need to calculate the area between the x-axis and the two curves: the parabola and the tangent line.
First, let's find the intersection points between the parabola and the tangent line. The given tangent line passes through the point (3, 45). We can substitute x = 3 in the equation of the parabola to find the corresponding y-coordinate:
y = 5x^2
y = 5 * 3^2
y = 5 * 9
y = 45
So, the tangent line intersects the parabola at the point (3, 45).
Now, let's find the x-coordinate of the second intersection point between the parabola and the x-axis. We know that when the parabola intersects the x-axis, y = 0. We can set y = 0 in the equation of the parabola and solve for x:
0 = 5x^2
0 = x^2
From this equation, we can see that the parabola intersects the x-axis at x = 0.
Therefore, the region bounded by the parabola, the tangent line, and the x-axis lies between x = 0 and x = 3.
To find the area between the curves, we need to find the integral of the difference between the parabola and the tangent line with respect to x over this interval. The equation of the tangent line at (3, 45) is obtained by finding the slope of the parabola at x = 3. Differentiating the equation of the parabola with respect to x gives us:
dy/dx = 10x
Substituting x = 3 into the derivative, we get the slope of the tangent line at x = 3:
m = 10 * 3
m = 30
So, the equation of the tangent line is:
y = mx + c
y = 30x + c
To find c, we use the point (3, 45):
45 = 30 * 3 + c
45 = 90 + c
c = 45 - 90
c = -45
Therefore, the equation of the tangent line is:
y = 30x - 45
Now, we can find the area between the curves by calculating the integral from x = 0 to x = 3 of the difference between the parabola and the tangent line:
Area = ∫[(5x^2) - (30x - 45)] dx, where the limits of integration are from 0 to 3.
By evaluating this integral, we can find the area of the region bounded by the parabola, the tangent line, and the x-axis.
To find the area of the region bounded by the parabola y = 5x^2, the tangent line at (3, 45), and the x-axis, we need to calculate the area between the parabola and the x-axis and subtract the area between the tangent line and the x-axis.
Step 1: Calculate the area under the parabola:
To find the area under the parabola y = 5x^2, we can integrate the function from the x-values where it intersects the x-axis. To find these x-values, we set y = 5x^2 equal to 0:
0 = 5x^2
Divide both sides by 5:
x^2 = 0
This equation is satisfied when x = 0.
So, the area under the parabola is given by the integral:
A1 = ∫[0, 3] 5x^2 dx
Step 2: Calculate the area under the tangent line:
The equation of the tangent line at (3, 45) can be found by taking the derivative of the parabola at x = 3, which gives us the slope of the tangent line. Using the power rule for differentiation, we have:
y' = d/dx (5x^2)
y' = 10x
Substituting x = 3 into y' gives us the slope of the tangent line at x = 3:
m = 10(3) = 30
The equation of the tangent line is y = mx + b, where m is the slope and b is the y-intercept. We already have m, and we can find b by substituting the coordinates of the point (3, 45):
45 = 30(3) + b
45 = 90 + b
b = -45
So the equation of the tangent line is y = 30x - 45.
To find the area under the tangent line, we integrate the function from the x-values where it intersects the x-axis. Again, we set y = 30x - 45 equal to 0:
0 = 30x - 45
30x = 45
Divide both sides by 30:
x = 1.5
So, the area under the tangent line is given by the integral:
A2 = ∫[0, 1.5] (30x - 45) dx
Step 3: Calculate the area bounded by the parabola, the tangent line, and the x-axis:
The area of the region bounded by the parabola, the tangent line, and the x-axis is given by the difference between A1 and A2:
A = A1 - A2
A = ∫[0, 3] 5x^2 dx - ∫[0, 1.5] (30x - 45) dx
Now you can calculate the definite integrals and subtract to find the area.
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