1. Calculate the Riemann sum for $\int_0^{12} f(x) dx$ using 5 right endpoint rectangles with points $x = 0,1,3,8,11,12$ and values $f(x) = 2,-2,-3,-2,-4,-5$. - Intervals: $[0,1], [1,3], [3,8], [8,11], [11,12]$ - Widths: $1, 2, 5, 3, 1$ - Heights (right endpoints): $f(1)=-2, f(3)=-3, f(8)=-2, f(11)=-4, f(12)=-5$ - Riemann sum = $1\times(-2) + 2\times(-3) + 5\times(-2) + 3\times(-4) + 1\times(-5) = -2 -6 -10 -12 -5 = -35$ 2. Calculate the Riemann sum for $\int_0^{12} f(x) dx$ using 4 right endpoint rectangles with points $x=0,2,6,9,12$ and $f(x)=-1,-2,-1,0,-1$. - Intervals: $[0,2], [2,6], [6,9], [9,12]$ - Widths: $2,4,3,3$ - Heights (right endpoints): $f(2)=-2, f(6)=-1, f(9)=0, f(12)=-1$ - Riemann sum = $2\times(-2) + 4\times(-1) + 3\times0 + 3\times(-1) = -4 -4 + 0 -3 = -11$ 3. Calculate the Riemann sum for $\int_0^{12} f(x) dx$ using 4 right endpoint rectangles with points $x=0,1,5,7,12$ and $f(x)=0,2,1,3,1$. - Intervals: $[0,1], [1,5], [5,7], [7,12]$ - Widths: $1,4,2,5$ - Heights (right endpoints): $f(1)=2, f(5)=1, f(7)=3, f(12)=1$ - Riemann sum = $1\times2 + 4\times1 + 2\times3 + 5\times1 = 2 + 4 + 6 + 5 = 17$ 4. Calculate the Riemann sum for $\int_0^{12} f(x) dx$ using 5 left endpoint rectangles with points $x=0,1,4,10,11,12$ and $f(x)=-1,-3,-2,2,4,4$. - Intervals: $[0,1], [1,4], [4,10], [10,11], [11,12]$ - Widths: $1,3,6,1,1$ - Heights (left endpoints): $f(0)=-1, f(1)=-3, f(4)=-2, f(10)=2, f(11)=4$ - Riemann sum = $1\times(-1) + 3\times(-3) + 6\times(-2) + 1\times2 + 1\times4 = -1 -9 -12 + 2 + 4 = -16$