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 = cube root(x) , 0 ≤ x ≤ 27
Isn't Wolfram neat ?
http://www.wolframalpha.com/input/?i=integral+cube+root%28x%29+dx+from+1+to+27
area = ∫x^(1/3) dx from 1 to 27
= (3/4)x^(4/3) from 1 to 27
= (3/4)( 81 - 1)
= 60
I must have entered my info in wrong cuz I was getting a really big number.
60 is incorrect it's not between 0 and 27
To estimate the area beneath the curve y = cube root(x), we can use a graph.
1. First, plot the curve y = cube root(x) on a coordinate system.
2. Select a number of equal intervals along the x-axis within the given range of x values (0 to 27 in this case).
3. Divide the area into these intervals and draw rectangles with the x-axis as their base and the curve as their top.
4. Calculate the area of each rectangle by multiplying its base (the width of the interval) by its height (the value of y at that point).
5. Sum up the areas of all the rectangles to get an estimate of the total area beneath the curve.
To find the exact area, we can use calculus.
1. The given curve is y = cube root(x). By integrating this function within the given range (0 to 27), we can find the exact area.
2. First, find the indefinite integral of the function by applying the power rule for integration: ∫x^(1/3) dx = (3/4) * x^(4/3) + C, where C is the constant of integration.
3. Evaluate the definite integral by plugging in the upper limit (27) and the lower limit (0) into the antiderivative obtained in the previous step: (∫ x^(1/3) dx) from 0 to 27 = [(3/4) * 27^(4/3)] - [(3/4) * 0^(4/3)].
4. Simplify the expression to get the exact area beneath the curve.
Using this method, you can estimate the area beneath the curve by using a graph and find the exact area by using calculus.