The interval \(\displaystyle [0, 8]\) divided into four sub-intervals gives rectangles with vertices of the bases at

\(\displaystyle x=0,2,4,6,8\)

For the Left Riemann sum, we need to find the rectangle heights which values come from the left-most function value of each sub-interval, or f(0), f(2), f(4), and f(6).

\(\displaystyle f(0) = 0+7 = 7\)

\(\displaystyle f(2) = 2+7 = 9\)

\(\displaystyle f(4) = 4+7 = 11\)

\(\displaystyle f(6) = 6+7 = 13\)

Because each sub-interval has a width of 2, the Left Riemann sum is

\(\displaystyle A = bh = 2(7+9+11+13) = 80\)

In order to find the Riemann Sum of a given function, we need to approximate the area under the line or curve resulting from the function using rectangles spaced along equal sub-intervals of a given interval. Since we have an interval \(\displaystyle [0,6]\) divided into \(\displaystyle 3\) sub-intervals, we'll be using rectangles with vertices at \(\displaystyle x=0,2,4,6\). 

To approximate the area under the curve, we need to find the areas of each rectangle in the sub-intervals. We already know the width or base of each rectangle is \(\displaystyle b=2\) because the rectangles are spaced \(\displaystyle 2\) units apart. Since we're looking for the Left Riemann Sum, we want to find the heights \(\displaystyle h\) of each rectangle by taking the values of each leftmost function value on each sub-interval, as follows:

\(\displaystyle f(0)=(0)^{2}=0\)

\(\displaystyle f(2)=(2)^{2}=4\)

\(\displaystyle f(4)=(4)^{2}=16\)

Putting it all together, the Left Riemann Sum is 

\(\displaystyle A=bh=2(0+4+16)=2(20)=40\).

In order to find the Riemann Sum of a given function, we need to approximate the area under the line or curve resulting from the function using rectangles spaced along equal sub-intervals of a given interval. Since we have an interval \(\displaystyle [0,4]\) divided into \(\displaystyle 4\) sub-intervals, we'll be using rectangles with vertices at \(\displaystyle x=0,1,2,3,4\). 

To approximate the area under the curve, we need to find the areas of each rectangle in the sub-intervals. We already know the width or base of each rectangle is \(\displaystyle b=1\) because the rectangles are spaced \(\displaystyle 1\) unit apart. Since we're looking for the Left Riemann Sum, we want to find the heights \(\displaystyle h\) of each rectangle by taking the values of each leftmost function value on each sub-interval, as follows:

\(\displaystyle f(0)=\frac{1}{4}(0)+2=0+2=2\)

\(\displaystyle f(1)=\frac{1}{4}(1)+2=\frac{1}{4}+\frac{8}{4}=\frac{9}{4}\)

\(\displaystyle f(2)=\frac{1}{4}(2)+2=\frac{1}{2}+\frac{4}{2}=\frac{5}{2}\)

\(\displaystyle f(3)=\frac{1}{4}(3)+2=\frac{3}{4}+\frac{8}{4}=\frac{11}{4}\)

Putting it all together, the Left Riemann Sum is 

\(\displaystyle A=bh=1(2+\frac{9}{4}+\frac{5}{2}+\frac{11}{4})=\frac{8}{4}+\frac{9}{4}+\frac{10}{4}+\frac{11}{4}=\frac{38}{4}=\frac{19}{2}\)

\(\displaystyle \begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{We're asked to approximate}\int_{3}^{3.9}(-9tan(10sin(5x)))dx\\&\text{So the interval is }[3,3.9]\text{ the subintervals have length }\frac{3.9-(3)}{3}=\frac{3}{10}\\&\text{and since we are using the *left*point of each interval, the x-values are:}\\&[3,\frac{33}{10},\frac{18}{5}]\\&\int_{3}^{3.9}(-9tan(10sin(5x)))dx=\frac{3}{10}[(-9tan(10sin(15)))+(-9tan(10sin(\frac{33}{2})))+(-9tan(10sin(18)))]\\&\int_{3}^{3.9}(-9tan(10sin(5x)))dx=9.91\end{align*}\)

\(\displaystyle \begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{We're asked to approximate}\int_{-2}^{11.6}(\frac{18}{3^{(4tan(4x))}})dx\\&\text{So the interval is }[-2,11.6]\text{ the subintervals have length }\frac{11.6-(-2)}{4}=\frac{17}{5}\\&\text{and since we are using the *left*point of each interval, the x-values are:}\\&[-2,\frac{7}{5},\frac{24}{5},\frac{41}{5}]\\&\int_{-2}^{11.6}(\frac{18}{3^{(4tan(4x))}})dx=\frac{17}{5}[(18\cdot 3^{(4tan(8))})+(\frac{18}{3^{(4tan(\frac{28}{5}))}})+(\frac{18}{3^{(4tan(\frac{96}{5}))}})+(\frac{18}{3^{(4tan(\frac{164}{5}))}})]\\&\int_{-2}^{11.6}(\frac{18}{3^{(4tan(4x))}})dx=2200.80\end{align*}\)

\(\displaystyle \begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{We're asked to approximate}\int_{-4}^{4.4}(6cos(11e^{(2x)}))dx\\&\text{So the interval is }[-4,4.4]\text{ the subintervals have length }\frac{4.4-(-4)}{3}=\frac{14}{5}\\&\text{and since we are using the *left*point of each interval, the x-values are:}\\&[-4,-\frac{6}{5},\frac{8}{5}]\\&\int_{-4}^{4.4}(6cos(11e^{(2x)}))dx=\frac{14}{5}[(6cos(11e^{(-8)}))+(6cos(11e^{(-\frac{12}{5})}))+(6cos(11e^{(\frac{16}{5})}))]\\&\int_{-4}^{4.4}(6cos(11e^{(2x)}))dx=41.86\end{align*}\)

\(\displaystyle \begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{We're asked to approximate}\int_{0}^{12}(11sin(20sin(x)))dx\\&\text{So the interval is }[0,12]\text{ the subintervals have length }\frac{12-(0)}{3}=4\\&\text{and since we are using the *left*point of each interval, the x-values are:}\\&[0,4,8]\\&\int_{0}^{12}(11sin(20sin(x)))dx=4[(0)+(11sin(20sin(4)))+(11sin(20sin(8)))]\\&\int_{0}^{12}(11sin(20sin(x)))dx=11.66\end{align*}\)

\(\displaystyle \begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{We're asked to approximate}\int_{0}^{3}(-16sin(2x^{2}))dx\\&\text{So the interval is }[0,3]\text{ the subintervals have length }\frac{3-(0)}{3}=1\\&\text{and since we are using the *left*point of each interval, the x-values are:}\\&[0,1,2]\\&\int_{0}^{3}(-16sin(2x^{2}))dx=1[(0)+(-16sin(2))+(-16sin(8))]\\&\int_{0}^{3}(-16sin(2x^{2}))dx=-30.38\end{align*}\)

\(\displaystyle \begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{We're asked to approximate}\int_{4}^{12.7}(-17cos(20x^{2}))dx\\&\text{So the interval is }[4,12.7]\text{ the subintervals have length }\frac{12.7-(4)}{3}=\frac{29}{10}\\&\text{and since we are using the *left*point of each interval, the x-values are:}\\&[4,\frac{69}{10},\frac{49}{5}]\\&\int_{4}^{12.7}(-17cos(20x^{2}))dx=\frac{29}{10}[(-17cos(320))+(-17cos(\frac{4761}{5}))+(-17cos(\frac{9604}{5}))]\\&\int_{4}^{12.7}(-17cos(20x^{2}))dx=16.39\end{align*}\)

\(\displaystyle \begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{We're asked to approximate}\int_{0}^{9}(-\frac{4}{4^e^{(6x)}})dx\\&\text{So the interval is }[0,9]\text{ the subintervals have length }\frac{9-(0)}{3}=3\\&\text{and since we are using the *left*point of each interval, the x-values are:}\\&[0,3,6]\\&\int_{0}^{9}(-\frac{4}{4^e^{(6x)}})dx=3[(-1)+(-\frac{4}{4^e^{(18)}})+(-\frac{4}{4^e^{(36)}})]\\&\int_{0}^{9}(-\frac{4}{4^e^{(6x)}})dx=-3.00\end{align*}\)