Snap chalk lines across the sidewalk just beyond the damaged area, keeping the line straight and square. Cut into the concrete with a concrete demolition saw, which is available at tool rental stores.
Break up the larger chunks with the sledgehammer until they are small enough to remove by hand. Remove all concrete debris from the area to be replaced. Enclose the emptied space with horizontal 2-by-4 lumber. Drive more 2-by-4 lumber vertically against the horizontal forms to stake it in place.
Remove 3 inches of soil from the area to be replaced, measuring from the bottom of the 2-by-4 forms. Place 3 inches of crusher-run gravel in the area and tamp it. Make sure that the gravel remains below the bottom of the 2-by-4 forms.
Mix the concrete and water in the wheelbarrow. Stir the concrete with a hoe until it is soupy. Add more water -- 1 gallon at a time -- if the mix is too dry.
Pour the concrete into the area to be replaced. Fill it to the top of the wooden forms and to the cut edges of the sidewalk. Smooth the concrete in a circular motion with the trowel, pressing the concrete gravel down into the concrete soup.
Allow the concrete to set for 30 minutes before proceeding. In temperatures below 40 degrees, the setting time may take up to 45 minutes to one hour, however.
Slide the concrete edger along the edges of the newly poured concrete where it meets the 2-by-4 forms to round off the edges. Slide the concrete jointer along the joints where the new concrete meets the old, following the line of old concrete. The jointer will separate the concrete slightly to prevent the two -- new and old -- from cracking.
Pull the broom across the surface of the concrete to achieve a sanded, broom finish. Do not push the broom, as this may damage the drying concrete. Do not place any pressure on the broom. Allow only the broom's weight to be applied to the surface.
Remove the wooden forms from the poured concrete and place the tarp over it. Leave the tarp in place for two days to prevent excess drying of the concrete.