The following functions are negative on the given interval....

The following functions are negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph off and the $x$ -axis on the interval using a left, right, and midpoint Riemann sum with $n=4$. $$f(x)=-4-x^{3} \text { on }[3,7]$$

The following functions are negative on the given interval.
a. Sketch the function on the given interval.
b. Approximate the net area bounded by the graph off and the $x$ -axis on the interval using a left, right, and midpoint Riemann sum with $n=4$.
$$f(x)=-4-x^{3} \text { on }[3,7]$$

Key Concepts

Riemann Sums

This concept involves approximating the area under a curve by summing the areas of a finite number of rectangles. In this method, the width of each subinterval is determined by dividing the total interval into equal parts, and the height of each rectangle is based on the function value at specific sample points. Variants include left endpoint, right endpoint, and midpoint evaluations, each providing different approximations of the overall area.

Definite Integrals and Net Area

The definite integral of a function over an interval represents the net area between the function’s graph and the x-axis, taking into account areas below the x-axis as negative and those above as positive. When the function is entirely negative over the interval, the computed integral is negative, but the actual area is considered as the magnitude of this value.

Subinterval Partitioning

This concept is crucial for setting up Riemann sums. It involves dividing the overall interval into a set number of equal subintervals, which simplifies the calculation of approximate areas. The width of each subinterval, denoted as ?x, is obtained by subtracting the lower limit from the upper limit and dividing by the number of subintervals.

Graph Sketching

Sketching a graph involves drawing an accurate visual representation of the function on a coordinate plane over a given interval. This process includes identifying key features of the function such as its shape, behavior at the boundaries, and any symmetry or monotonicity. Understanding the sign of the function is important since it affects the interpretation of the area under the curve relative to the x-axis.

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