The area is given exactly by the definite integral,
We can write this as a Riemann sum, i.e. the infinite sum of rectangular areas:
• Split up the integration interval into <em>n</em> equally-spaced subintervals, each with length (2 - (-3))/<em>n</em> = 5/<em>n</em> - - this will be the width of each rectangle. The intervals would then be
[-3, -3 + 5/<em>n</em>], [-3 + 5/<em>n</em>, -3 + 10/<em>n</em>], …, [-3 + 5(<em>n</em> - 1)/<em>n</em>, 2]
• Over each subinterval, take the function value at some point <em>x</em> * to be the height.
Then the area is given by
Now, if we take <em>x</em> * to be the left endpoint of each subinterval, we have
<em>x</em> * = -3 + 5(<em>k</em> - 1)/<em>n</em> → <em>f</em> (<em>x</em> *) = (-3 + 5(<em>k</em> - 1)/<em>n</em>)² + 4
If we instead take <em>x</em> * to be the right endpoint, then
<em>x</em> * = -3 + 5<em>k</em>/<em>n</em> → <em>f</em> (<em>x</em> *) = (-3 + 5<em>k</em>/<em>n</em>)² + 4
So as a Riemann sum, the area is represented by
and if you expand the summand, this is the same as
So from the given choices, the correct ones are
• row 1, column 1
• row 2, column 2
• row 4, column 2