Question Video: Finding the Area under a Curve Using Sigma Notation with Riemann Sums Mathematics • Higher Education

Video Transcript

Represent the area under the curve of the function 𝑓 of 𝑥 equals 𝑥 squared minus one on the close interval zero to three in sigma notation using a right Riemann sum with 𝑛 subintervals.

Remember when we’re finding a right Riemann sum, we find the sum of 𝛥𝑥 times 𝑓 of 𝑥 𝑖 for values of 𝑖 from one to 𝑛. 𝛥𝑥 is 𝑏 minus 𝑎 over 𝑛 where 𝑎 and 𝑏 are the lower and upper limits of our interval, respectively, and 𝑛 is the number of subintervals. 𝑥 𝑖 is 𝑎 plus 𝑖 lots of 𝛥𝑥. We always begin by working out what 𝛥𝑥 is. In our case, 𝑎 is equal to zero, 𝑏 is equal to three, and, well, 𝑛 is just 𝑛. This means 𝛥𝑥 is three minus zero over 𝑛 or just three over 𝑛. Next, we’re going to work out what 𝑥 𝑖 is. It’s 𝑎, which we know to be zero, plus 𝛥𝑥, which is three over 𝑛, times 𝑖. We’ll write this as three 𝑖 over 𝑛.

Of course, we want to know what 𝑓 of 𝑥 𝑖 is. So it follows that to find 𝑓 of 𝑥 𝑖, we find 𝑓 of three 𝑖 over 𝑛. Let’s substitute three 𝑖 over 𝑛 into our formula. That’s three 𝑖 over 𝑛 all squared minus one which is nine 𝑖 squared over 𝑛 squared minus one. We’re now ready to use the summation formula. We’re evaluating our sum for values of 𝑖 from one to 𝑛. Its 𝛥𝑥, which is three over 𝑛, multiplied by nine 𝑖 squared over 𝑛 squared minus one. We distribute our parentheses and then we’re going to look to create a common denominator. We can do that by multiplying both the numerator and denominator of our second fraction by 𝑛 squared. That gives us three 𝑛 squared over 𝑛 cubed, leaving us just to simply combine the numerators. We have 27 𝑖 squared minus three 𝑛 squared over 𝑛 cubed.

Now, we can simplify this somewhat. The numerators share a factor of 27 and three. And of course they have a common denominator of 𝑛 cubed. Both three and 𝑛 cubed are indepnent of 𝑖. This means we can take three over 𝑛 cubed outside of the sigma symbol, and that means we’re done. We’ve represented the area under the curve of the function in sigma notation with a right Riemann sum. It’s three over 𝑛 cubed times the sum of nine 𝑖 squared minus 𝑛 squared for values of 𝑖 from one 𝑛.