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 = 6 sin x, 0 ≤ x ≤ π
We use integration.
∫ y dx
= ∫ 6 sin(x) dx
= -6 cos(x) | from 0 to π
= -6 cos(π) - (-6 cos(0))
= -6(-1) - (-6)(1)
= 6 + 6
= 12
How are you supposed to know that you need to use integration to do this?
Because he said the area below the graph of the function he provided.
And Yes my name is Nael.
Oh, I'm sorry, but I can't actually draw graphs. However, I can try my best to help you estimate the area beneath the curve!
Alright, let's break it down. The function y = 6 sin x represents a sine wave with an amplitude of 6. When x = 0, y = 0, and when x = π, y = 6 sin π = 0.
Since the graph is contained in the interval 0 ≤ x ≤ π, we just need to find the positive area beneath the curve.
Now, the exact area can be found by calculating the definite integral of the function over the given interval. In this case, the integral of y = 6 sin x with respect to x from 0 to π is:
∫(0 to π) 6 sin x dx
Applying the integral, we get:
-6(cos x) | (0 to π)
Plugging in the limits, we have:
-6(cos π) - (-6(cos 0))
Since cos π = -1 and cos 0 = 1, the expression simplifies to:
-6(-1) - (-6(1)) = 6 + 6 = 12
So, the exact area beneath the curve y = 6 sin x from 0 to π is 12.
Keep in mind that this is just an estimate, and it's always better to use calculus for precise calculations. But hey, at least I tried to bring a little humor to the world of math!
To estimate the area of the region that lies beneath the curve y = 6 sin(x) for the given interval 0 ≤ x ≤ π, we can create a rough graph of the curve on a coordinate plane.
First, let's draw the x-axis and the y-axis on the plane. The x-axis will represent the x-values within the interval, and the y-axis will represent the corresponding y-values.
Next, we plot points on the graph by choosing several x-values within the given interval and finding their corresponding y-values by evaluating y = 6 sin(x) for each x-value. The more points we plot, the better the estimate will be.
For example, we can choose x = 0, x = π/4, x = π/2, x = 3π/4, and x = π as our x-values. Evaluating y = 6 sin(x) for each of these x-values, we get the following y-values:
- For x = 0, y = 6 sin(0) = 0.
- For x = π/4, y = 6 sin(π/4) ≈ 4.24.
- For x = π/2, y = 6 sin(π/2) ≈ 6.
- For x = 3π/4, y = 6 sin(3π/4) ≈ 4.24.
- For x = π, y = 6 sin(π) = 0.
Now, we plot these points on the graph.
After plotting the points, we connect them with a smooth curve. This curve will be an approximation of the shape of the graph y = 6 sin(x) within the given interval.
To estimate the area, we can count the number of complete squares and partial squares "underneath" the curve. Each square on the graph paper represents a unit area.
In this case, it seems that the region underneath the curve resembles a rectangle with a triangular segment on top. We can count the number of complete squares and partial squares within this region.
To find the exact area, we need to use integral calculus to evaluate the definite integral of y = 6 sin(x) from x = 0 to x = π:
Exact Area = ∫[0, π] 6 sin(x) dx.
By evaluating this definite integral, we can find the exact area of the region that lies beneath the curve y = 6 sin(x) for the given interval 0 ≤ x ≤ π.
Keep in mind that this explanation is a rough estimate using a graph, and to find the exact area, you need to use calculus techniques.