WEBVTT
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Estimate the integral from zero to one of two times 𝑒 to the power of 𝑥 squared plus three 𝑥 with respect to 𝑥 using the trapezoidal rule with five subintervals.
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Round your answer to one decimal place.
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The question is asking us to estimate this integral by using the trapezoidal rule.
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And we recall we can estimate the definite integral from 𝑎 to 𝑏 of a function 𝑓 of 𝑥 with respect to 𝑥 by using the trapezoidal rule with 𝑛 subintervals.
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As Δ𝑥 over two multiplied by 𝑓 evaluated at 𝑥 zero plus 𝑓 evaluated at 𝑥 𝑛 plus two times 𝑓 evaluated at 𝑥 one plus 𝑓 evaluated at 𝑥 two.
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And we add this all the way up to 𝑓 evaluated at 𝑥 𝑛 minus one.
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Well, Δ𝑥 is equal to 𝑏 minus 𝑎 over 𝑛.
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And each of our 𝑥 𝑖 are equal to 𝑎 plus 𝑖 times Δ𝑥.
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We get this approximation since the definite integral is an area.
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And we can approximate this area by using 𝑛 trapezoids of width Δ𝑥.
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Since we want to use the trapezoidal rule with five subintervals, we’ll set our value of 𝑛 equal to five.
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We see the lower limit of our integral is zero and the upper limit is one.
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So we’ll set our value of 𝑎 equal to zero and our value of 𝑏 equal to one.
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Finally, we’ll set our function 𝑓 of 𝑥 equal to our integrand two times 𝑒 to the power of 𝑥 squared plus three 𝑥.
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The first thing we want to do is calculate the value of Δ𝑥.
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We have that Δ𝑥 is equal to 𝑏 minus 𝑎 over 𝑛.
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We know that 𝑏 is equal to one, 𝑎 is equal to zero, and 𝑛 is equal to five.
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So Δ𝑥 is equal to one minus zero over five, which is equal to one-fifth or 0.2.
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So we’ve found the value of Δ𝑥.
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We’ve found the first part of our trapezoidal formula.
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We now need to calculate 𝑓 evaluated at each of our values of 𝑥 𝑖.
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To help us find these values, we’ll make a table containing the values of 𝑖, 𝑥 𝑖, and 𝑓 evaluated at 𝑥 𝑖.
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Since 𝑛 is equal to five, we need to find the value of 𝑥 five.
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We also need to find the value of 𝑥 four, 𝑥 three, 𝑥 two, 𝑥 one, and 𝑥 zero.
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So we’ll need six columns for our values of 𝑖.
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In fact, we’ll always need one more column than the value of 𝑛 for our values of 𝑖.
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To find our values of 𝑥 𝑖, we recall that 𝑥 𝑖 is equal to 𝑎 plus 𝑖 times Δ𝑥.
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Since 𝑎 is equal to zero and Δ𝑥 is equal to 0.2, this gives us that 𝑥 𝑖 is equal to zero plus 𝑖 times 0.2, which simplifies to give us 0.2 times 𝑖.
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We can then substitute the values of 𝑖 into this equation to find the values for 𝑥 𝑖.
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Substituting 𝑖 is equal to zero gives us that 𝑥 zero is equal to 0.2 times zero, which is zero.
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Substituting 𝑖 is equal to one gives us that 𝑥 one is equal to 0.2 times one, which is 0.2.
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We can do the same to find the rest of our values of 𝑥 𝑖.
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We have 𝑥 two is equal to 0.4, 𝑥 three is equal to 0.6, 𝑥 four is equal to 0.8, and 𝑥 five is equal to one.
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Now we need to find our function 𝑓 evaluated at each of these values of 𝑥 𝑖.
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We recall that 𝑓 of 𝑥 is equal to our integrand two times 𝑒 to the power of 𝑥 squared plus three 𝑥.
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Substituting 𝑥 𝑖 is equal to zero, we get two times 𝑒 to the power of zero squared plus three times zero.
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And our exponent of zero squared plus three times zero simplifies to give us zero.
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And 𝑒 to the power of zero is just equal to one.
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So 𝑓 evaluated at 𝑥 zero is just equal to two.
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We can do the same for 𝑥 one.
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We substitute 0.2 into our function 𝑓 of 𝑥, to give us two times 𝑒 to the power of 0.2 squared plus three times 0.2.
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And again, we can evaluate the expression in our exponent.
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This time, we get an exponent of 0.64.
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So we have 𝑓 evaluated at 𝑥 one is two times 𝑒 to the power of 0.64.
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We can do the same to find 𝑓 evaluated at the rest of our values of 𝑥 𝑖.
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We get 𝑓 evaluated at 𝑥 two is two times 𝑒 to the power of 1.36.
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𝑓 evaluated at 𝑥 three is two times 𝑒 to the power of 2.16.
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𝑓 evaluated at 𝑥 four is two times 𝑒 to the power of 3.04.
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And 𝑓 evaluated at 𝑥 five is two times 𝑒 to the fourth power.
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We’ve now calculated all of the values we need to estimate our integral by using the trapezoidal rule with five subintervals.
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We’ve shown that Δ𝑥 is equal to 0.2.
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So Δ𝑥 over two is equal to 0.2 divided by two.
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Since 𝑛 is equal to five, we’ve shown that 𝑓 evaluated at 𝑥 zero and 𝑓 evaluated at 𝑥 five are equal to two and two times 𝑒 to the fourth power, respectively.
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Then we need to add two times 𝑓 evaluated at 𝑥 one plus 𝑓 evaluated at 𝑥 two.
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And we add all the way up to 𝑓 evaluated at 𝑥 four.
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And we calculated 𝑓 evaluated at 𝑥 one, two, three, and four in our table.
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So we just need to calculate this expression.
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And remember, the question wants us to give our answer to one decimal place.
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So we need to calculate the value of this expression to one decimal place.
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And we get 25.3.
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Therefore, by using the trapezoidal rule with five subintervals, we’ve shown that the integral from zero to one of two times 𝑒 to the power of 𝑥 squared plus three 𝑥 with respect to 𝑥 is approximately equal to 25.3.