Video Transcript
Estimate the integral between two and three of the square root of 𝑥 cubed add two 𝑥 minus four with respect to 𝑥 using the trapezoidal rule with four subintervals. Approximate your answer to two decimal places.
Let’s begin by writing out the formula for the trapezoidal rule. And for this formula, Δ𝑥 equals 𝑏 minus 𝑎 over 𝑛, where 𝑛 is the number of subintervals. 𝑥 zero to 𝑥 𝑛 are the end points of the trapezoids that we used to estimate the area under the curve.
Let’s start by finding our Δ𝑥 using the formula Δ𝑥 equals 𝑏 minus 𝑎 over 𝑛, 𝑎 and 𝑏 the limits of integration. So this is three minus two over four because we’re using four subintervals. This is one over four or 0.25. So if we have four subintervals with width 0.25 between two and three, we can calculate the values of 𝑥 zero, 𝑥 one, 𝑥 two, 𝑥 three, and 𝑥 four to be two, 2.25, 2.5, 2.75, and three.
Note that, alternatively, we could’ve used the formula 𝑥𝑖 equals 𝑎 add 𝑖Δ𝑥 for 𝑖 from zero to 𝑛. So let’s now write out our formula for the trapezoidal rule with Δ𝑥 that we found and the values of 𝑥 that we found. From here, we can calculate these values of 𝑓 of different values of 𝑥.
If we start with 𝑓 of two, this is the function inside the integral, the square root of 𝑥 cubed add to 𝑥 minus four, evaluated at two. We find that this gives us 2.828. And we do the same for 2.25, 2.5, 2.75, and three. And we substitute these values into our working. From here, we have that 0.25 over two is 0.125. 2.828 add 5.385 is 8.213. 3.448 add 4.077 add 4.722 is 12.247. So this gives us 0.125 multiplied by 8.213 add two multiplied by 12.247. And so we find our approximation to be 4.09 to two decimal places.
