Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.
y = 5th root(x), 0 ≤ x ≤ 32
Nevermind. I figured this one out.
Use integration to get the exact area:
∫ y dx
∫ x^(1/5) dx
Recall that
∫ (x^n) dx = 1/(n+1) x^(n+1)
Thus, for example,
∫ (x^2) dx = (1/3)x^3
Try this method on ∫ x^(1/5) dx , and evaluate it from 0 to 32.
To find a rough estimate of the area under the curve, you can use a numerical method called the Left Riemann Sum. This involves dividing the interval [0, 32] into a certain number of small subintervals and approximating the area of each subinterval as a rectangle.
Here's how you can determine a rough estimate using a graph:
1. Start by drawing the coordinate axes on a graph paper, marking the x-axis from 0 to 32 and the y-axis from 0 to the maximum value of y for the given curve.
2. Divide the x-axis into a few equal subintervals (e.g., 4 or 6) by drawing vertical lines at equally spaced intervals.
3. At each division point on the x-axis, draw a rectangle with its height equal to the function value at that point and its width equal to the width of the subinterval.
4. Add up the areas of all the rectangles to get an estimate of the total area under the curve.
This rough estimate will give you an idea of the approximate area, but it won't be precise. To find the exact area, you can use calculus and integrate the function over the given interval.
In this case, the function is y = 5th root(x). To find the exact area, you need to evaluate the definite integral of this function from 0 to 32:
∫(0 to 32) 5th root(x) dx
To evaluate this integral, you can use integration techniques such as substitution or the power rule.
By integrating the function, you will find the exact area under the curve between x = 0 and x = 32.